Advanced statistical mechanics

INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS SERIES EDITORS J. BIRMAN S. F. EDWARDS R. FRIEND M. REES D. SHERRINGTON G...

Author: Barry M McCoy

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INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS SERIES EDITORS J. BIRMAN S. F. EDWARDS R. FRIEND M. REES D. SHERRINGTON G. VENEZIANO

CITY UNIVERSITY OF NEW YORK UNIVERSITY OF CAMBRIDGE UNIVERSITY OF CAMBRIDGE UNIVERSITY OF CAMBRIDGE UNIVERSITY OF OXFORD CERN, GENEVA

International Series of Monographs on Physics 146. B. McCoy: Advanced statistical mechanics 145. M. Bordag, G.L. Klimchitskaya, U. Mohideen, V.M. Mostepanenko: Advances in the Casimir effect 144. T.R. Field: Electromagnetic scattering from random media 143. W. G¨ otze: Complex dynamics of glass-forming liquids - a mode-coupling theory 142. V.M. Agranovich: Excitations in organic solids 141. W.T. Grandy: Entropy and the time evolution of macroscopic systems 140. M. Alcubierre: Introduction to 3+1 numerical relativity 139. A. L. Ivanov, S. G. Tikhodeev: Problems of condensed matter physics - quantum coherence phenomena in electron-hole and coupled matter-light systems 138. I. M. Vardavas, F. W. Taylor: Radiation and climate 137. A. F. Borghesani: Ions and electrons in liquid helium 136. C. Kiefer: Quantum gravity, Second edition 135. V. Fortov, I. Iakubov, A. Khrapak: Physics of strongly coupled plasma 134. G. Fredrickson: The equilibrium theory of inhomogeneous polymers 133. H. Suhl: Relaxation processes in micromagnetics 132. J. Terning: Modern supersymmetry 131. M. Mari˜ no: Chern-Simons theory, matrix models, and topological strings 130. V. Gantmakher: Electrons and disorder in solids 129. W. Barford: Electronic and optical properties of conjugated polymers 128. R. E. Raab, O. L. de Lange: Multipole theory in electromagnetism 127. A. Larkin, A. Varlamov: Theory of fluctuations in superconductors 126. P. Goldbart, N. Goldenfeld, D. Sherrington: Stealing the gold 125. S. Atzeni, J. Meyer-ter-Vehn: The physics of inertial fusion 123. T. Fujimoto: Plasma spectroscopy 122. K. Fujikawa, H. Suzuki: Path integrals and quantum anomalies 121. T. Giamarchi: Quantum physics in one dimension 120. M. Warner, E. Terentjev: Liquid crystal elastomers 119. L. Jacak, P. Sitko, K. Wieczorek, A. Wojs: Quantum Hall systems 118. J. Wesson: Tokamaks, Third edition 117. G. Volovik: The Universe in a helium droplet 116. L. Pitaevskii, S. Stringari: Bose-Einstein condensation 115. G. Dissertori, I.G. Knowles, M. Schmelling: Quantum chromodynamics 114. B. DeWitt: The global approach to quantum field theory 113. J. Zinn-Justin: Quantum field theory and critical phenomena, Fourth edition 112. R.M. Mazo: Brownian motion - fluctuations, dynamics, and applications 111. H. Nishimori: Statistical physics of spin glasses and information processing - an introduction 110. N.B. Kopnin: Theory of nonequilibrium superconductivity 109. A. Aharoni: Introduction to the theory of ferromagnetism, Second edition 108. R. Dobbs: Helium three 107. R. Wigmans: Calorimetry 106. J. K¨ ubler: Theory of itinerant electron magnetism 105. Y. Kuramoto, Y. Kitaoka: Dynamics of heavy electrons 104. D. Bardin, G. Passarino: The Standard Model in the making 103. G.C. Branco, L. Lavoura, J.P. Silva: CP Violation 102. T.C. Choy: Effective medium theory 101. H. Araki: Mathematical theory of quantum fields 100. L. M. Pismen: Vortices in nonlinear fields 99. L. Mestel: Stellar magnetism 98. K. H. Bennemann: Nonlinear optics in metals 94. S. Chikazumi: Physics of ferromagnetism 91. R. A. Bertlmann: Anomalies in quantum field theory 90. P. K. Gosh: Ion traps 87. P. S. Joshi: Global aspects in gravitation and cosmology 86. E. R. Pike, S. Sarkar: The quantum theory of radiation 83. P. G. de Gennes, J. Prost: The physics of liquid crystals 73. M. Doi, S. F. Edwards: The theory of polymer dynamics 69. S. Chandrasekhar: The mathematical theory of black holes 51. C. Møller: The theory of relativity 46. H. E. Stanley: Introduction to phase transitions and critical phenomena 32. A. Abragam: Principles of nuclear magnetism 27. P. A. M. Dirac: Principles of quantum mechanics 23. R. E. Peierls: Quantum theory of solids

Advanced Statistical Mechanics Barry M. McCoy CN Yang Institute for Theoretical Physics State University of New York Stony Brook, NY

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Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c Barry McCoy 2010 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2010 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Printed in Great Britain on acid-free paper by CPI Antony Rowe, Chippenham, Wiltshire ISBN 978–0–19–955663–2 (Hbk.) 1 3 5 7 9 10 8 6 4 2

Preface The best way to become acquainted with a subject is to write a book about it. Benjamin Disraeli The subject of statistical mechanics dates back to the 19th century. It has a rich history, and the basics of the subject are taught in all undergraduate and graduate physics programs. Consequently there is a wealth of books that explain the elementary aspects of the subject which form the foundation for all thermal properties of condensed matter systems. The content of these books is all rather similar in that they cover thermodynamics, ensemble theory, one-body problems and the perfect Bose and Fermi gases. These topics are all considered to be closed subjects which are thoroughly understood. This book is an outgrowth of the author’s teaching of advanced courses in statistical mechanics which go beyond the topics covered in elementary courses and is aimed at introducing the reader to topics in which there is ongoing research. In contrast to the material in an elementary course almost all topics lead to open questions, and the aim of this book is to present these topics of ongoing research to as wide an audience as possible. Consequently in almost all chapters there are sections on open questions and what I call missing theorems where one’s physical intuition suggests that results should be true but for which no proof yet exists. It is hoped that, by highlighting the many places where there are unresolved questions, this book can stimulate progress in the field. The selection of topics in any advanced treatment of a subject is affected by the tastes of the author and so several comments about my selection of topics are in order. I have chosen to divide the subject somewhat arbitrarily into three parts: exact general theorems; series expansions and numerical results; and solvable models. Each of these divisions has an immense literature and within the confines of one book it is not possible to state all results and prove all known theorems. I have therefore adopted the procedure of stating and explaining many results but have only given the proofs of a selection of the theorems stated. There is no other alternative since there are many important theorems whose proof in the literature requires papers of 40–50 pages. For example the proof of the stability of matter is the subject of a book in its own right by Elliot Lieb; Rodney Baxter devotes an entire book to the free energy and order parameters of the six-vertex, eight-vertex and hard hexagon models; T.T. Wu and the present author devote an entire book to the Ising model. In this sense this present book can be considered to be an introduction and guide to, but is hardly a substitute for, the literature of the past 50 years.

Preface

The reader will almost instantly note that there are several well-known topics in statistical physics which are not covered in this book: namely the renormalization group and mean field theory. This omission is deliberate since both of these topics are well covered in many books and it is, in my opinion, superfluous to give one more account of these methods. Constant progress is being made in statistical mechanics and it is certain that even at the time of publication some topics will have advanced beyond what is presented here. In particular I draw the reader’s attention to several examples of recent work: the proof discussed in chapter 4 of Kepler’s conjecture that no packing of hard spheres in three dimensions can be more dense than the face centered cubic lattice, and the discovery also presented in chapter 4 that for ellipsoids there are packings denser than the fcc lattice. There are several computations presented which were initiated in part by the desire to clarify and better understand the existing literature, in particular the computation of the tenth order virial coefficients in chapter 7, the diagonal susceptibility and the evaluation of the form factor integrals of the Ising model in chapter 10 and 12 and the treatment of the TQ equation of the eight-vertex model in chapter 14. There are also many interesting and important problems which are omitted merely for lack of space. In particular there are many solvable models which have not been mentioned. Furthermore there is no discussion of the methods of the coordinate and algebraic Bethe’s ansatz and the mathematics of quantum groups and conformal field theory. These topics require much more space than this book allows and are treated extensively by other authors. However, it is hoped that in spite of the many necessary omissions that this book covers a sufficiently large number of topics so that the reader will gain an appreciation of the great breadth of the subject; the many areas of progress which have been made in the past 40–50 years, and the places where future advances will be made. I am fond of saying that “you cannot say that you understand a paper until you generalize it.” This, of course, leads to the logical corollary that “no author can be said to understand his/her most recent paper.” This book is an excellent demonstration of the truth and meaning of this corollary. In every chapter there are open questions and topics that need further research and explanation. Some of the more obvious and unavoidable of these questions have been singled out for discussion but many, if not most, are quietly hidden away waiting for the reader to discover them. There are many derivations and computations presented and, barring misprints, the proofs should be sufficient to prove the conclusions. But in no place is it ever shown that the given proof is actually necessary for the conclusion and that the steps exhibited actually reveal the mechanism for the phenomena being discussed. The most glaring example of this problem is the evaluation of integrals done in chapter 12 by the use of MAPLE for which, at the time of writing, no analytic derivation exists. There are many places in this book where I need to thank collaborators and friends for their help and suggestions: Nathan Clisby for the evaluations of virial coefficients in chapter 7; Jean-Marie Maillard for teaching me how to do the symbolic evaluation of integrals on the computer in chapter 12; Klaus Fabricius for collaboration on the Q matrices of the eight-vertex model of chapter 14; and Jacques Perk and Helen Au-Yang for collaboration on chiral Potts models and for figures 4 and 5 of chapter 15. The

Preface

method used in chapter 6 to prove the Mayer expansion comes from a set of lectures given by Hans Groeneveld in the late 1960s who has given me much valuable advice in the preparation of that chapter. I am most grateful to the National Science Foundation for partial support during much of the time when I was writing this book and to the Rockefeller Foundation for a one-month residency at their Bellagio Conference and Study Center where several chapters were revised and perfected. In conclusion I must thank and acknowledge two remarkable people to whom I am deeply indebted and without whose encouragement and inspiration this book would never have been completed. The first is my late wife, Tun-Hsu Martha McCoy , who helped me every step of the way and put up with the innumerable frustrations I have had during the far too many years I have spent in writing. The other is my classmate of 51 years ago from Catalina High School in Tucson, Arizona, Margaret Hagen Wright, who has given me profound friendship in a time of great need. Stony Brook, New York 2009

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Contents PART I GENERAL THEORY 1

Basic principles 1.1 Thermodynamics 1.1.1 Macroscopic, extensive and intensive 1.1.2 Equilibrium 1.1.3 The four laws of thermodynamics 1.2 Statistical mechanics 1.2.1 Statistical philosophy 1.2.2 The microcanonical ensemble 1.2.3 The canonical ensemble 1.2.4 The grand canonical ensemble 1.2.5 Phases and ergodic components 1.3 Quantum statistical mechanics 1.3.1 The relation of classical to quantum statistical mechanics 1.4 Quantum field theory References

3 3 3 5 6 9 9 10 11 15 17 17 18 19 21

2

Reductionism, phenomena and models 2.1 Reductionism 2.2 Phenomena 2.2.1 Monatomic insulators 2.2.2 Diatomic insulators 2.2.3 Liquid crystals 2.2.4 Water 2.2.5 Metals 2.2.6 Helium 2.2.7 Magnetic transitions 2.3 Models 2.3.1 Continuum models 2.3.2 Lattice models 2.4 Discussion 2.5 Appendix: Bravais lattices References

22 22 24 24 25 28 28 29 29 30 33 34 37 41 42 44

3

Stability, existence and uniqueness 3.1 Classical stability 3.1.1 Catastrophic potentials 3.1.2 Conditions for stability 3.1.3 Superstability

45 49 49 49 57

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Contents

3.1.4 Multispecies interactions Quantum stability 3.2.1 Stability of matter 3.2.2 Proofs of theorems 1 and 2 3.3 Existence and uniqueness of the thermodynamic limit 3.3.1 Box boundary conditions 3.3.2 Periodic boundary conditions 3.3.3 Existence and uniqueness in the canonical ensemble 3.3.4 Existence and uniqueness in the grand canonical ensemble 3.3.5 Continuity of the pressure 3.4 First order phase transitions, zeros and analyticity 3.5 Discussion 3.6 Open questions 3.7 Appendix A: Properties of functions of positive type 3.8 Appendix B: Fourier transforms References

59 61 61 63 66 67 69 69 77 78 80 82 84 85 86 90

Theorems on order 4.1 Densest packing of hard spheres and ellipsoids 4.2 Lack of order in the isotropic Heisenberg model in D = 1, 2 4.3 Lack of crystalline order in D = 1, 2 4.4 Existence of ferromagnetic and antiferromagnetic order in the classical Heisenberg model (n vector model) in D = 3 4.4.1 The mechanism for ferromagnetic order 4.4.2 Proof of the bound (4.123) 4.4.3 Antiferromagnetism 4.5 Existence of antiferromagnetic order in the quantum Heisenberg model for T > 0 and D = 3 4.6 Existence of antiferromagnetic order in the quantum Heisenberg model for T = 0 and D = 2 4.7 Missing theorems References

92 93 97 103

3.2

4

5

Critical phenomena and scaling theory 5.1 Thermodynamic critical exponents and inequalities for Ising-like systems 5.2 Scaling theory for Ising-like systems 5.2.1 Scaling for H = 0 5.2.2 Scaling for H = 0 5.2.3 Summary of critical exponent equalities 5.3 Scaling for general systems 5.3.1 The classical n vector and quantum Heisenberg models 5.3.2 Lennard-Jones fluids 5.4 Universality 5.5 Missing theorems References

110 111 113 117 118 120 120 122 124 125 128 129 132 136 136 137 142 142 143 145

Contents

Ü

PART II SERIES AND NUMERICAL METHODS 6

Mayer virial expansions and Groeneveld’s theorems 6.1 The second virial coefficient 6.2 Mayers’ first theorem 6.3 Mayers’ second theorem 6.3.1 Step 1 6.3.2 Step 2 6.3.3 Step 3 6.4 Non-negative potentials and Groeneveld’s theorems 6.5 Convergence of virial expansions 6.6 Counting of Mayer graphs 6.7 Appendix: The irreducible Mayer graphs of four and five points References

149 156 158 160 160 162 164 167 173 176 178 180

7

Ree–Hoover virial expansion and hard particles 7.1 The Ree–Hoover expansion 7.2 The Tonks Gas 7.3 Hard sphere virial coefficients B2 –B4 in two and higher dimensions 7.3.1 Evaluation of B2 7.3.2 Evaluation of B3 7.3.3 Evaluation of B4 7.4 Monte-Carlo evaluations of B5 –B10 7.5 Hard sphere virial coefficients for k ≥ 11 7.6 Radius of convergence and approximate equations of state 7.7 Parallel hard squares, parallel hard cubes and hard hexagons on a lattice 7.8 Convex nonspherical hard particles 7.9 Open questions References

181 182 186

High density expansions 8.1 Molecular dynamics 8.2 Hard spheres and discs 8.2.1 Behavior near close packing 8.2.2 Freezing of hard spheres 8.2.3 The phase transition for hard discs 8.3 The inverse power law potential 8.3.1 Scaling behavior 8.3.2 Numerical computations 8.4 Hard spheres with an additional square well 8.5 Lennard-Jones potentials 8.6 Conclusions References

210 211 212 213 214 219 222 223 224 225 227 228 230

8

189 189 191 194 195 196 198 202 204 205 208

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Contents

9

High temperature expansions for magnets at H = 0 9.1 Classical n vector model for D = 2, 3 9.1.1 Results for D = 2 9.1.2 A qualitative interpretation of the D = 2 data 9.1.3 Results for D = 3 9.1.4 Critical exponents 9.1.5 The ratio method 9.1.6 Estimates from differential approximates 9.2 Quantum Heisenberg model 9.2.1 Results for D = 2 9.2.2 Results for D = 3 9.2.3 Analysis of results 9.3 Discussion 9.4 Statistical mechanics versus quantum field theory 9.5 Appendix: The expansion coefficients for the susceptibility on the square lattice References PART III

232 234 237 240 242 243 248 254 255 257 258 259 261 265 267 272

EXACTLY SOLVABLE MODELS

10 The Ising model in two dimensions: summary of results 10.1 The homogeneous lattice at H = 0 10.1.1 Partition function on the torus 10.1.2 Zeros of the partition function 10.1.3 Bulk free energy per site 10.1.4 Partition function at T = Tc 10.1.5 Spontaneous magnetization 10.1.6 Row and diagonal spin correlation functions 10.1.7 The correlation C(M, N ) for general M, N 10.1.8 Scaling limit 10.1.9 Magnetic susceptibility of the bulk 10.1.10 The diagonal susceptibility 10.2 Boundary properties of the homogeneous lattice at H = 0 10.2.1 Boundary free energy at Hb = 0 10.2.2 Boundary magnetization M1 (Hb ) 10.2.3 Boundary spin correlations 10.2.4 Analytic continuation and hysteresis 10.3 The layered random lattice 10.4 The Ising model for H = 0 10.4.1 The circle theorem 10.4.2 The imaginary magnetic field H/kB T = iπ/2 10.4.3 Expansions for small H 10.4.4 T = Tc with H > 0 10.4.5 Extended analyticity References

277 280 280 281 283 286 286 287 295 297 302 306 309 309 310 312 314 316 319 319 319 321 322 323 324

Contents

Ü

11 The Pfaffian solution of the Ising model 11.1 Dimers 11.1.1 Dimers on lattices with free boundary conditions 11.1.2 Dimers on a cylinder 11.1.3 Dimers on lattices of genus g ≥ 1 11.1.4 Explicit evaluation of the Pfaffians 11.1.5 Thermodynamic limit 11.1.6 Other lattices and boundary conditions 11.2 The Ising partition function 11.2.1 Toroidal (periodic) boundary conditions 11.2.2 Cylindrical boundary conditions 11.3 Correlation functions 11.3.1 The correlation σM,N σM,N 11.3.2 The diagonal correlation σ0,0 σN,N 11.3.3 Correlations near the boundary References

328 329 330 337 338 339 344 345 347 347 354 355 355 359 360 361

12 Ising model spontaneous magnetization and form factors 12.1 Wiener–Hopf sum equations 12.1.1 Fourier transforms 12.1.2 Splitting and factorization 12.1.3 Solution 12.2 Spontaneous magnetization and Szeg¨ o’s theorem 12.2.1 Proof of Szeg¨ o’s theorem 12.2.2 The spontaneous magnetization 12.3 Form factor expansions of C(N, N ) and C(0, N ) 12.3.1 Expansion for T < Tc 12.3.2 Expansion for T > Tc 12.4 Asymptotic expansions of C(N, N ) and C(0, N ) for N → ∞ 12.4.1 Large N for T < Tc 12.4.2 Large N for T > Tc 12.4.3 Large N for T = Tc 12.5 Evaluation of diagonal form factor integrals 12.5.1 Differential equations 12.5.2 Factorization and direct sums 12.5.3 Homomorphisms of operators 12.5.4 Symmetric powers 12.5.5 Results 12.5.6 Discussion References

363 364 364 366 367 368 369 374 375 375 386 392 392 393 393 398 399 399 402 402 404 406 407

13 The star–triangle (Yang–Baxter) equation 13.1 Historical overview 13.2 Transfer matrices 13.2.1 Explicit forms of the transfer matrix 13.2.2 The physical regime

408 408 412 415 416

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Contents

13.3 Integrability 13.4 Star–triangle equation for vertex models 13.4.1 Boltzmann weights for two-state vertex models 13.4.2 Vertex–spin correspondence 13.4.3 Inhomogeneous lattices 13.5 Star–triangle equation for spin models 13.5.1 Chiral Potts model 13.5.2 Proof of the star–triangle equation 13.5.3 Determination of Rpqr 13.6 Star–triangle equation for face models 13.6.1 SOS and RSOS models 13.6.2 The hard hexagon model 13.7 Hamiltonian limits 13.7.1 Spin chains for the eight- and six-vertex models 13.7.2 Spin chain for the chiral Potts model 13.8 Appendix: Properties of theta functions References

417 418 419 436 439 440 440 448 451 452 452 457 464 466 469 472 477

14 The eight-vertex and XYZ model 14.1 Historical overview 14.2 The matrix TQ equation for the eight-vertex model 14.2.1 Modified theta functions 14.2.2 Formal construction of the matrices Q72 (v) 14.2.3 Explicit construction of QR (v) and QL (v) 14.2.4 The interchange relation 14.2.5 Nonsingularity and nondegeneracy 14.2.6 Quasiperiodicity 14.3 Eigenvalues and free energy 14.3.1 The form of the eigenvalues 14.3.2 Numerical study of the eigenvalues of Q72 (v) 14.3.3 Bethe’s equation 14.3.4 Computation of the free energy 14.4 Excitations, order parameters and correlation functions of the eight- and six-vertex model 14.4.1 Eight-vertex polarization P8 and XYZ order 14.4.2 Eight-vertex magnetization M8 14.4.3 Correlations for the XY model 14.4.4 XYZ correlations 14.5 Appendix: Properties of the modified theta functions References

480 481 484 486 488 489 498 508 509 514 514 516 526 528 537 538 541 542 550 552 557

15 The hard hexagon, RSOS and chiral Potts models 15.1 The hard hexagon and RSOS models 15.1.1 Historical overview 15.1.2 Hard hexagons for 0 ≤ z ≤ zc 15.1.3 Hard hexagons for zc ≤ z < ∞

562 562 562 565 571

Contents

15.1.4 Discussion 15.2 The chiral Potts model 15.2.1 Historical overview 15.2.2 Real and positive Boltzmann weights 15.2.3 The superintegrable chiral Potts model and Onsager’s algebra 15.2.4 The functional equation for the superintegrable case for N =3 15.2.5 Superintegrable ground state energy for small λ 15.2.6 Single particle excitations and level crossing 15.2.7 Order parameter 15.2.8 The phase diagram of the spin chain 15.3 Open questions 15.3.1 Q operators 15.3.2 Degenerate subspaces for the eight-vertex model 15.3.3 Symmetry algebra for the eight-vertex model at roots of unity 15.3.4 Chiral Potts correlations References

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574 575 575 578 585 588 590 595 598 599 600 601 602 603 603 605

PART IV CONCLUSION 16 Reductionism versus complexity 16.1 Does history matter? 16.2 Size is important 16.3 The paradox of integrability 16.4 Conclusion References

613 613 615 616 617 618

Index

619


Part I General Theory

A child of five could understand this. Fetch me a child of five. Groucho Marx


1 Basic principles This book is called Advanced Statistical Mechanics and as such it is assumed that the reader has studied thermodynamics and knows the ensemble formulation of statistical mechanics. The derivation of these principles from classical mechanics and the application of these laws to free and simple one-body systems is covered in previous courses at both the undergraduate and graduate level. This course is devoted to applying these principles to the study of interacting many-body systems to derive properties of macroscopic matter from microscopic interactions. However, in order to have a common starting point for our investigations we will in this chapter summarize the laws and results of thermodynamics and present the formulation of ensemble theory in the form which it will be used in this course.

1.1

Thermodynamics

The study of large systems originates with the investigations of the 19th century into the subject of thermodynamics, a subject of great importance for the development of steam power which was at the heart of the industrial revolution. The laws of thermodynamics rule our daily lives and the thermodynamic notions of heat, temperature and efficiency are as common as the weather report and the kitchen stove. And yet for all of its ubiquitous truth thermodynamics is a strange subject. Its invention in the 19th century did not develop logically from Newton’s laws of motion. Rather it seems to be a “final cause” in the sense of Aristotle. We are forced to study thermodynamics because nature empirically turns out to work this way. Because thermodynamic behavior exists it must follow from the microscopic laws of nature, regardless of how much we may not like it. Indeed, the late 19th century discussions of various “paradoxes” connected with Poincaré recurrence and irreversibility demonstrate that there were many who did not seem to like thermodynamics one bit. In this section we will sketch these laws of thermodynamics. Their microscopic justification can only be said to have been completed in the late 20th century. The phenomena of thermodynamics form the simplest and best understood properties of large systems. 1.1.1

Macroscopic, extensive and intensive

Thermodynamics begins with a definition of the words macroscopic, extensive and intensive. By macroscopic we mean that for a system of N particles in a volume V we are interested in the behavior when

Basic principles

N → ∞ and V → ∞ with V /N = v fixed

(1.1)

This limit is called the thermodynamic limit, v is called the specific volume, and 1/v = ρ is called the (number) density. The words extensive and intensive refer to the behavior of properties of this N -body system in the thermodynamic limit. A property is called intensive if it is independent of N in the limit (1.1) and is called extensive if it is linear in N . More precisely we have lim intensive = constant (1.2) N →∞

and lim extensive/N = constant

N →∞

(1.3)

and these limits are independent of the shape of the system. Thermodynamics assumes that there are no other ways that a property can depend on the number of particles. In particular, if thermodynamics is applicable to a system, its total energy must be extensive. But this assertion that only intensive and extensive properties exist already imposes a limitation on the type of microscopic interactions we are allowed to deal with. In particular we must be able to guarantee that the system will neither collapse because of short distance attraction nor explode because of long distance repulsion. The problem with short distance attraction is seen by considering the familiar case of a collection of N hard spheres of radius r and mass m where the interaction between two spheres of different masses separated by a distance d is the gravitational interaction Gm1 m2 V (d) = . (1.4) d To compute the potential energy of a spherical collection of these small spheres consider adding one sphere to a spherical collection of N spheres. If we let R denote the radius of the composite sphere, then the dependence of R on N is found by computing the volume of the collection of N spheres (4/3)πR3 = constant N (4/3)πr3

(1.5)

where the constant is geometrically determined by closest packing. Thus for large N R ∼ N 1/3

(1.6)

From (1.4) and (1.6) we see that the potential energy gained in adding one sphere to the collection of N spheres is Gm(mN ) Gm2 N ∼ ∼ N 2/3 R N 1/3

(1.7)

and hence if we integrate from one to N we find that the average gravitational potential energy of the N closest packed gravitating hard spheres is proportional to N 5/3 which is neither intensive nor extensive. Therefore we conclude that the early 19th century

Thermodynamics

laws of thermodynamics will not apply to gravity which was the only force which had been mathematically investigated at the time that thermodynamics was first invented! The possibility of a large system exploding in the thermodynamic limit is most dramatically seen in the construction of an atomic bomb. Here there is a critical mass beyond which an uncontrolled chain reaction sets in and there is no sense in which equilibrium describes the physics. A slightly less drastic situation is the case of a system with a net charge. Here an electrostatic charge will reside on the surface of the system and the total energy will depend on the shape of the system. The conditions on the potential which allow intensive and extensive behavior were not really understood until the mid 20th century, and it was not until the 1960s, when it was shown that the ground state energy of a system of electrons interacting with a neutralizing positive charge background is linear in N , that there was a proof that any realistic set of microscopic forces actually will have the dependence on the number of particles assumed by thermodynamics. We will study these questions in detail in chapter 3. 1.1.2

Equilibrium

The most “obvious” empirical property of macroscopic matter is that if we isolate it (for example by putting it in a Dewar) and wait long enough all changes in the system will die out. This is to be contrasted with a system of a few degrees of freedom such as two or three billiard balls on a frictionless table which will keep on bouncing forever. Much of statistical mechanics deals with the computations of properties of this equilibrium state. However, the very statement of what is meant by equilibrium seems to fly in the face of the first law of motion. Consequently it is of great importance to understand where the concept comes from. To use the words proof or derivation is far too strong. The entire development of classical statistical mechanics can be said to be part of the proof of the existence of equilibrium. The notion of equilibrium is in essence a statement that for a very large system there are several time scales. One of these time scales is finite when the number of particles goes to infinity. This scale may be on the order of microseconds or hours but once this time scale is reached the extensive and intensive properties of the large system cease to change. However, there may be other much longer time scales which become infinite as the size goes to infinity and on these extremely large scales change may again occur. Thermal equilibrium refers to the time after these finite time scales have been surpassed but the infinite ones have not yet been reached. This statement of waiting long enough is admittedly vague because there are systems like glass which change slowly over a period of several hundreds of years. Consequently it is often useful to give an alternative definition and to say that something is in equilibrium if we can describe all of its properties that are subject to macroscopic measurements in terms of a few intensive and extensive variables (such as pressure and density) which do not depend on the past history of the system. These few variables needed to characterize the system are said to specify its state. Properties that depend only on the state of the system and not on the past history are called state functions. The independence of past history is clearly a genuine restriction on the types of phe-

Basic principles

nomena that can be discussed. Thus the discussion of equilibrium properties marks only the beginning of the study of the properties of large systems. The dramatic feature of equilibrium is that it is an irreversible concept which puts into physics a direction of time even if the underlying microscopic equations are time reversal invariant. One of the first tasks of microscopic statistical mechanics is to demonstrate the existence of these phenomena. 1.1.3

The four laws of thermodynamics

Thermodynamics is embodied in the following four laws which are abstracted from our observations of macroscopic matter in equilibrium: Zeroth Law: If, of three bodies A, B and C, the bodies A and B are separately in equilibrium with C then A and B are in equilibrium with each other. First Law: If the state of an otherwise isolated system is changed by the performance of work, the amount of work needed depends solely on the change accomplished and not on the means by which the work is performed, or on the intermediate stages through which the system passes between its initial and final states. Second Law: (Clausius) It is impossible to devise an engine which, working in a cycle, shall produce no effect other than the transfer of heat from a colder to a hotter body; (Kelvin) It is impossible to devise an engine which, working in a cycle, shall produce no effect other than the extraction of heat from a reservoir and the performance of an equal amount of work. Third Law: (Nernst) The entropy change associated with any isothermal, reversible process approaches zero as T → 0; (Fowler and Guggenheim) It is impossible by any procedure to reduce any system to absolute zero in a finite number of operations. The first three of these four laws have a purely classical origin and each leads to the existence of a new state function. The zeroth law allows us to define the state function called empirical temperature (denoted by φ) which has the property that any two bodies in equilibrium have the same value of the empirical temperature. This empirical temperature defines a set of isotherms relative to one (arbitrarily) chosen standard system called a thermometer. The first law lets us define the state function called internal energy. The change in this internal energy in going from one state to another is the work added to the system in thermal isolation in changing the state. The typical experimental demonstration of this is the Joule paddle wheel experiment in which the stirring of a mass of water raises the temperature. Once this state function is defined we can consider the more general situation where the change of state occurs without the system being thermally isolated. For example, by holding a Bunsen burner under a flask we can change the state without adding any mechanical work at all. In this general situation the change in internal energy consists of two terms; the work done on the system and the heat, ∆Q, transferred to the system. In symbols ∆U = ∆W + ∆Q.

(1.8)

Thermodynamics

Moreover, in connection with the first law we recognize that there are two distinct ways to change the state of a system. One way is to perform the change so slowly that equilibrium is maintained at all times. At any time we can reverse the direction of this change and thus such quasistatic changes are called reversible. The slow pulling out of a piston where ∆W = −P dV (1.9) exemplifies this type of change. On the other hand we could suddenly expand a gas into a free volume by breaking a membrane. Now only the initial and final states of the system are in equilibrium. Such changes cannot be reversed in time and are called irreversible. The second law allows the definition of a state function, S, called entropy that describes the change in heat during a reversible change. In particular in a reversible change ∆Q = dQ = T dS,

(1.10)

dU = T dS − P dV .

(1.11)

and hence Here T , called the absolute temperature, is a function of the empirical temperature alone and, if the thermometer is chosen as defined from the perfect gas law P v = kB T

(1.12)

where kB = 1.38 × 10−23 Joules/degree Kelvin is called Boltzmann’s constant, then the absolute and the empirical scales of temperature agree. This calibration of thermometers to agree with the perfect gas temperature is clearly a difficult task to do at very low temperatures where there are no good approximations to a perfect gas. Nevertheless great experimental ingenuity has gone into the design of thermometers that make experimental measurements in terms of absolute temperature with great accuracy and we will always unite the concepts of empirical and absolute temperature in this book and speak of temperature alone. These state functions are measurable and of great importance in studying macroscopic systems. One of the most common of measurements is of the extensive quantity C called the heat capacity, defined as the change in the heat with respect to temperature at some fixed external conditions. For example, in a gas specified by the state variable P and V there are two commonly measured heat capacities: dQ dQ Cv = and Cp = (1.13) dT V dT P It is also common to divide by the number of particle N and consider the specific heats cv = Cv /N and cp = Cp /N.

(1.14)

The computation of specific heats will be one of the major topics of this book. It remains to consider the third law. This was only discovered in the 20th century and stems from quantum instead of classical mechanics. Its major consequence for us

Basic principles

can be seen if we combine the definition of the entropy state function (1.10) with the definition of heat capacity (1.13) to write dS = C(T )dT /T. Thus

S(T2 ) − S(T1 ) =

(1.15)

T2

dT C(T )/T.

(1.16)

T1

The third law says that as T1 → 0 this entropy change goes to a constant and thus we see that the heat capacity of any system must vanish as T → 0. The rate at which the heat capacity vanishes gives important information about the quantum interactions of the system. The derivation of these state functions from the laws, and the relations between experimental quantities that can be derived from these laws, requires space and attention to detail. In particular we will find it useful at times to use the quantities 1. Helmholtz free energy A = U − T S where dA = −P dV − SdT

(1.17)

G = U − T S + P V where dG = V dP − SdT

(1.18)

H = U + P V where dH = T dS + V dP,

(1.19)

2. Gibbs function

3. Enthalpy

the four relations of Maxwell

∂T ∂V S ∂T ∂P S ∂V ∂T P ∂P ∂T V

∂P =− ∂S V ∂V = ∂S P ∂S =− ∂P T ∂S = ∂V T

(1.20)

and the internal energy of the perfect gas U=

3N kB T. 2

(1.21)

An excellent set of derivations is given in the classic book The Elements of Classical Thermodynamics by A.B. Pippard [1].

Statistical mechanics

1.2


Classical statistical mechanics was invented at the end of the 19th century by Maxwell, Boltzmann and Gibbs as a way to compute the phenomena of thermal equilibrium from the microscopic laws of classical mechanics. However, if the study of thermal equilibrium required a detailed knowledge of the solution of Hamilton’s equations even our knowledge in late 20th century would be insufficient. The reason that statistical mechanics could be founded long before such things as the KAM theorem were known is precisely because the relevant mechanics is statistical. 1.2.1

Statistical philosophy

From the beginning of philosophical inquiry 2400 years ago there has been a persistent dichotomy of thought as to what constitutes reality. On the one hand there are the empiricists who, starting with Aristotle, maintain that there exists an external reality which we learn of by perception. On the other hand there are the idealists who, starting with Plato, maintain that reality consists of mental ideas which we carry in our heads. An understanding of the relation between these two extreme points of view is necessary for an understanding of the relation of statistics to mechanics. The fundamental point in all philosophic and scientific investigations is the distinction between the observer and observed, and the relationship they have with each other. This relationship is clarified if we distinguish three situations in terms of information content: 1. The observer has infinitely more information capacity than the observed. 2. The observer and the observed have comparable information capacity. 3. The observed has infinitely more information capacity than the observer. It is the case 1), where the observer has infinitely more capacity to measure and store information than the observed object has degrees of freedom, that we most commonly think of when we consider Newtonian planetary motion or more generally any mechanical system described by a few degrees of freedom in the laboratory. In this case we have large and intricate devices for measuring positions and velocities and a great deal of capacity for storing the results of these measurements. Indeed the first major advances in astronomy were due to the ability to store and correlate data using a time period of over 1000 years starting with data taken in 720 B.C. in Mesopotamia. As our ability to make measurements increases in accuracy we are able to make use of the classical laws of motion to make predictions ever further into the future. This is the situation envisioned in the philosophy of empiricism. The limiting case of an infinite specification of the observed object is what the empirical philosophy defines as “physical reality.”, Such infinite specification can only be done if the observer has infinite information capacity compared to the observed object. Case 2) listed above is the situation which we commonly meet in ordinary human affairs. In our everyday dealings with people our knowledge is inevitably incomplete and nevertheless we are forced to act on the basis of what we actually know. While we can and often do use a fiction that there is an “objective reality” in practice we never

½¼

Basic principles

know what it is. The manner in which we predict future human actions is dominated by the severe limitations on our observations of the external world. The final case 3), where the observed object has infinitely more degrees of freedom than the observer has information capacity to measure, is the subject of statistical mechanics. In this case the idealist philosophy of Plato is indispensable because even if one wants to believe in an immutable external empirical world it is forever inaccessible. The inescapable Platonic idea that must be introduced into the study of infinite classical systems by finite means is the idea of a density of points in phase space ρ(pj , qj , t). This density is a mandatory concept because by definition we can never specify an individual point. To repeat: The density of points in phase space is a pure Platonic idea. It exists in the mind only and cannot be measured empirically. The object of statistical mechanics is to use these densities to make predictions of properties of infinite systems using only finite means. 1.2.2

The microcanonical ensemble

The initial question to be asked about the prediction of properties of infinite systems using only finite means is whether there are any finite means possible that will let us predict anything at all. This was the question asked by Maxwell, Boltzmann and Gibbs more than a century ago. They argued that, for a system whose energy is conserved, we at least know the value of the conserved energy. Then, if we knew absolutely nothing else about the system, we would be forced to say that we were studying a density which was constant on the surface of constant energy. This density function is called the microcanonical ensemble: ρMC ({pj .qj }) = δ(H({pj , qj }) − E)/Ω(E) where

(1.22)

d2N {pj , qj }δ(H({pj , qj }) − E).

Ω(E) =

(1.23)

The normalizing factor Ω(E) is called the structure function. However, it might be objected that not even the total energy can in principle be known exactly and hence we might consider as an alternative ρMC ({pj , qj }) = 1/Ω(E; ∆) for E ≤ H ≤ E + ∆, where

(1.24)

d2N {pj .qj }.

Ω(E) = E≤H≤E+∆

We will refer to both of these as the microcanonical ensemble.

(1.25)


½½

Maxwell, Boltzmann and Gibbs conjectured that the value of any macroscopic property A of a system with 2N degrees of freedom in thermal equilibrium is given in terms of this microcanonical density as A = d2N {pj , qj }ρA. (1.26) On the assumption that this ensemble and average formula do describe thermal equilibrium we will shortly find expressions for the thermodynamic state functions temperature, internal energy and entropy in terms of the structure function. This will demonstrate the existence of a formalism that allows the computations of nontrivial thermodynamic properties in terms of a microscopic Hamiltonian. This is done for a system of N identical particles in a volume V by identifying the thermodynamic entropy in terms of the structure function as S(E, V ) = kB ln [Ω(E, V )/N !] ,

(1.27)

from which, by using(1.11), we identify the absolute temperature as 1 ∂S(E, V ) = . T ∂E

(1.28)

To prove that this identification of the entropy is correct we must establish two key properties: 1. The limit where N → ∞, V → ∞ with V /N − v fixed lim

N →∞

1 S exists N

(1.29)

and 2. The temperatures of any two systems in thermal contact are equal. The proof of these properties depends on the Hamiltonian of the system. The conditions on the potential which allow these properties will be discussed in detail in chapter 3. 1.2.3

The canonical ensemble

In the presentation of the microcanonical ensemble the typical experimental situation under consideration is one where a macroscopic system approaches equilibrium in thermal isolation from its surroundings. In practice, however, this is not the most typical experimental configuration. It is much more common to have the macroscopic system on which we are making measurements placed in thermal contact with an even larger macroscopic system (the heat bath) whose function is to keep the observed system in thermal equilibrium. In this experimental situation we have absolutely no interest in the dynamics of the heat bath whose only function is to define the temperature. We have no interest in discussing the question of how the heat bath itself approaches equilibrium. The only property of the heat bath that we will use is the empirical observation that heat baths exist.

½¾

Basic principles

We are only interested in the observed system and not the heat bath and thus we are free to describe the heat bath by any fiction we please just so long as thermal equilibrium is maintained. The most convenient way to do this is to let H be the Hamiltonian of the system under observation and to represent the heat bath by Nh systems each with the identical Hamiltonians Hj . Thus the total Hamiltonian of the system plus the heat bath is Htot = H +

Nh

Hj .

(1.30)

j=1

In the limit when Nh → ∞ this represents the definition of a heat bath as having infinitely more degrees of freedom than the observed system. We represent the statement that the observed system is in thermal equilibrium with the heat bath by applying the microcanonical ensemble to the entire system (1.30) with the microcanonical density function ρMC = δEtot ,Htot /Ω(Etot ). (1.31) To study the system H we need the density function with all the coordinates of the heat bath integrated out. Thus we consider Etot ) = dX (1) · · · dX (N ) ρMC ρ(X, (1.32) h are the coordinates of H and X (j) are the coordinates of Hj . In the limit where X Nh → ∞ the ensemble represented by this density is called the canonical ensemble. We evaluate the limit Nh → ∞ by use of the method of steepest descents. To carry this out we will slightly simplify the formalism and assume that all energies are a common multiple of a unit ∆E. (Nothing will be lost in this argument since in the end nothing will depend on ∆E anyway.) We use this convention to write the Kronecker δ in (1.31) as π 1 θ(j − k) δj,k = . (1.33) dθ exp i 2π −π ∆E Thus, defining ζ = iθ/∆E, the denominator of (1.31) is written as   π Nh iθ 1 X (1) · · · dX (N ) (Etot − H − Ω(Etot ) = dXd dθ exp  Hj ) 2π −π ∆E j=1 ∆E = 2πi

iπ/∆E

dζ e(ζEtot +ln Z(ζ)+Nh ln Zh (ζ))

(1.34)

−iπ/∆E

where we have defined Z(ζ) =

−ζH and Zh = dXe

and the numerator of (1.32) is

(j) e−ζHj , dX

(1.35)


(1) · · · dX (N ) 1 dX h 2π =

½¿

 Nh iθ (Etot − H − dθ exp  Hj ) ∆E −π j=1



π

∆E 2πi

πi/∆E

dζe(ζ(Etot −H)+Nh ln Zh (ζ))

(1.36)

−πi/∆E

Consider first the structure function (1.34) in the limit Nh → ∞, Etot → ∞ with ¯ fixed and write the integrand as Etot /Nh = E ¯

Z(ζ)eNh [ζ E+ln Zh (ζ)] .

(1.37)

The steepest descents point maximizes the value of the exponential as a function of ζ and is found as the solution of ¯ + ∂ ln Zh (ζ) = 0. E ∂ζ

(1.38)

We denote the solution of this equation as β which is easily shown to be real and positive. Then, deforming the contour of integration to the steepest descents path of constant phase which passes through β and on this path setting ζ = β + iy we find to leading order in Nh that ∆E ∞ − N2h y2 ∂ 2 ln∂βZ2h (β) ¯ Ω(Etot ) ∼ Z(β)eNh [β E+ln Zh (β)] e 2π −∞ −1/2

∂ 2 ln Zh (β) 1 ¯ Nh [β E+ln Z(β)] 1 + O( = Z(β)e ∆E 2πNh ) . (1.39) ∂β 2 Nh A similar steepest descents evaluation of the numerator (1.36) gives −1/2

∂ 2 ln Zh (β) 1 ¯ 1 + O( e−βH eNh [β E+ln Zh (β)] ∆E 2πNh ) . ∂β 2 Nh

(1.40)

Therefore in the Nh → ∞ limit all the dependence on Nh , Zh and ∆E cancels out of (1.32) and we find that the resulting limiting density is ρC = e−βH /Z(β).

(1.41)

This is the density of the canonical ensemble and the normalizing factor Z(β) is called the partition function. With this density the expectation value of any observable f is given as e−βH /Z(β). f = dXf (1.42) In particular, the internal energy is ∂ ln Z(β) −βH . /Z(β) = − U = H = dXHe ∂β

(1.43)

It now remains to relate the canonical ensemble to thermodynamics. First note that, by construction, if H1 and H2 are two different systems in contact with the

½

Basic principles

same heat bath they will both have a canonical density given by (1.41) with the same value of β. Therefore β is the same for any two systems in thermal equilibrium and thus, from the zeroth law, β must be a universal function of temperature. To find this dependence it thus suffices to consider the perfect gas 1 2 p 2m j=1 j N

H=

(1.44)

confined in a volume V. For this system Z(β) =

3N

d

3N

xd

pe

1 −β 2m

N j=1

p2j

=V

N

2πm β

3N/2 (1.45)

and thus from (1.43) the internal energy is U=

3N . 2β

(1.46)

Thus comparing with the internal energy of the perfect gas obtained from thermodynamics (1.21) 3N U= kB T (1.47) 2 we obtain the general relation 1 . (1.48) β= kB T To complete the relation with thermodynamics we need to express the Helmholtz free energy in terms of Z(β). In particular we consider a system with N identical particles. In order to obtain a free energy which is extensive we set ZN (β) = Z(β)/N !

(1.49)

and define the Helmholtz free energy A(V, T ) from ZN (β) = e−βA(V,T ) .

(1.50)

N ! ∼ N N +1/2 e−N (2π)1/2

(1.51)

Using Stirling’s approximation

we see from (1.45) that, for the perfect gas, lim

V →∞, N →∞ v=V /N

1 lnZN V

(1.52)

with v fixed exists. We also find from (1.43) that U=

∂ ∂ βA(V, T ) = A(V, T ) − T A(V, T ) ∂β ∂T

(1.53)


from which, if we recall from thermodynamics (1.17) that ∂A ∂A and P = − , S=− ∂T V ∂V T

½

(1.54)

we conclude that the definition (1.50) does indeed satisfy all the required properties of the Helmholtz free energy. In the canonical ensemble the energy is not precisely defined. Only the average energy is computable. But for a large system the probability that the true energy is far from the energy is very small. As a measure of the fluctuation consider the average

2 variance H 2 − H . On the one hand ∂U ∂ −βH dXHe = /Z(β) ∂β ∂β

2 2 −βH −βH /Z(β) + /Z(β) = − dXH e XHe = − H 2 + H2 (1.55) and on the other hand from the thermodynamic definition of specific heat ∂U ∂U = kT 2 = kT 2 N cv − ∂β ∂T V

(1.56)

and thus

1 2 2 H = kT 2 cv /N. − H (1.57) N2 The two terms on the left are each of order one but the right-hand side is of order 1/N as long at the specific heat cv is finite in the thermodynamic limit. Thus we see that the fluctuations of the energy from the average value vanish as N → ∞. This is an indication that large chaotic systems can have very predictable average quantities and is an illustration of the operation of the laws of large numbers in statistical mechanics. 1.2.4

The grand canonical ensemble

In order to use the canonical ensemble we must, in principle, compute the partition function ZN (β) for a system with a finite number of particles N in a finite volume V and take the limit N → ∞, V → ∞ with N/V =

1 = ρ fixed. v

(1.58)

However, there are situations where it is technically more convenient to consider, instead of the partition function ZN (β), the grand partition function Qgr (z) =

∞

z N ZN (β).

(1.59)

N =0

In the grand canonical ensemble thermodynamic functions are expressed directly in terms of Qgr (z).

½

Basic principles

To express the thermodynamic functions in terms the grand partition function we use Cauchy’s theorem to write 1 ZN (β) = 2πi

dz z N +1

Qgr (z),

(1.60)

where the contour of integration is around z = 0. We study ZN (β) in the limit (1.58) by first writing (1.60) as ZN (β) =

1 2πi

dz [ln Qgr (z)−N ln z] e z

(1.61)

and then using the method of steepest descents. The steepest descents point is at ∂ [ln Qgr (z) − N ln z] = 0 ∂z

(1.62)

which, under the assumption that ln Qgr (z) is proportional to V, becomes in the limit (1.58) 1 ∂ z ln Qgr (z) = ρ. lim (1.63) V →∞ V ∂z The value of z which solves this equation is often written as z = eβµ and µ is called the chemical potential. We thus find ZN (β) ∼ e

[ln Qgr (z)−(N +1) ln z]

−1/2 ∂ 2 ln Qgr (z) 2π ∂z 2

(1.64)

and from (1.50) and (1.54) P =−

∂ ∂ [−β −1 ln ZN (β)] ∼ β −1 [ln Qgr (z) − (N + 1) ln z]. ∂V ∂V

(1.65)

Thus recalling that ln Qgr (z) depends linearly on V we may use ∂ 1 ln Qgr (z) = ln Qgr (z) ∂V V

(1.66)

and the fact that z is independent of V to obtain the desired result P 1 = lim ln Qgr (z). V →∞ V kB T

(1.67)

Finally we also obtain from (1.53) ∂ U 1 =− lim ln Qgr (z). V →∞ V ∂β V →∞ V lim

(1.68)

Quantum statistical mechanics

1.2.5

½

Phases and ergodic components

The microcanonical ensemble as presented above assumes that the only conservation law possessed by the system is the conservation of energy and therefore the microcanonical ensemble averages over all states with a given energy. More precisely the microcanonical ensemble assumes that for almost all initial conditions the system dynamically evolves in time such that eventually the system will come arbitrarily close to any given point on the surface of constant energy. This is called the “ergodic hypothesis” and has been studied for well over a century. The assumption that the energy is the only conserved quantity of a physical system in a large but finite box is technically possible because the boundary conditions of the box will break the translational invariance of the system. In the thermodynamic limit, however, boundary conditions should be irrelevant and thus we should be able to contemplate situations where the system is translationally and even rotationally invariant. For such systems with translational and/or rotational invariance there are more absolute constants of the motion besides the energy, and consequently the phase space decomposes into ergodic components which are characterized by representations of these invariance groups. Because of the invariance symmetry group the dynamical evolution of the system is restricted to a single ergodic component. These components which are characterized by different symmetry groups are said to be different pure phases of the system. If the representation is isotropic and homogeneous the system will be in a fluid phase but if the system has frozen into a crystal the allowed configurations will all have a crystalline symmetry and the only way that the system can pass from one phase into another is by means of interactions with the walls of the box. Therefore if we considered periodic instead of box boundary conditions each separate symmetry representation would have its own separate partition function and in order to find the “true” state of the system we would need to compute the partition function for all possible symmetry classes and then choose the free energy which was minimum. This partition function with the minimum free energy will on general depend on the temperature and a passage from say a face centered cubic to a body centered cubic phase will occur as a function of temperature when there is a crossing from one free energy to another. This is the phenomena of a first order phase transition. It cannot be seen by merely considering one phase in infinite space and can only be seen by comparing free energies of the different possible symmetry groups. A discussion of the concept of different ergodic components in infinite space is given by Ruelle [2] and a detailed discussion of the relation of symmetry groups to phases and phase transitions is given by Landau and Lifshitz [3]. We will return to this topic in chapters 3 and 4 when we discuss theorems on the existence of thermodynamic limits and on crystalline order.

1.3

Quantum statistical mechanics

The classical statistical mechanics presented in the previous section is a powerful tool in the study of macroscopic phenomena starting from microscopic interactions. However, the ultimate laws of nature which describe microscopic interactions are not classical

½

Basic principles

but are quantum mechanical. Therefore we must extend our statistical considerations to quantum statistics. Quantum mechanics has many differences from classical mechanics. One of the most basic is that the operators of position qj and of momentum pj of a particle do not commute but instead satisfy the commutation relation [qj , pk ] = i¯ hδj,k .

(1.69)

The consequence of this is that pj and qj cannot be simultaneously diagonalized and thus we cannot use the concept of orbit in phase space. But the concept of these orbits was basic to our classical understanding of the statistical philosophy that leads to the microcanonical ensemble. A second fundamental difference between classical and quantum mechanics is that in a finite size system the quantum energy levels are discrete whereas the classical energy is continuous. This discreteness of energy levels is the very reason for the word “quantum.” A major consequence of this discreteness is that the time behavior of any state or operator will be oscillatory and thus it is not to be expected that we can find ergodic behavior in the long time development of finite quantum systems. This is in great contrast to classical systems. Indeed there is no logical derivation of quantum statistical mechanics from first principles. Thus instead of attempting to derive a quantum version of the microcanonical and canonical ensemble from still underived “first principles” we will here state the rule which is used for computing thermal averages in quantum statistical mechanics. Quantum canonical ensemble: The quantum thermal average of any operator A is given as

with

A = TrAρ

(1.70)

ρ = e−βH /Z(β) Z(β) = Tre−βH

(1.71)

where β = 1/kB T and, by Tr, we mean the trace over all the states of the finite quantum system. This formula is the most obvious quantum mechanical extension of the canonical ensemble of classical mechanics. Indeed it is the only such extension known. However, owing to our incomplete understanding of the energy levels and eigenfunctions of large nonintegrable quantum systems we confess that we do not have any microscopic quantum theory from which this rule can be derived in a logically compelling manner. 1.3.1

The relation of classical to quantum statistical mechanics

Bulk matter can be described by nuclei and electrons interacting by means of the Coulomb interaction and nonrelativistic quantum mechanics. The stability theory of this system will be presented in chapter 3 but in actual practice this theory is not very tractable for concrete computations. To obtain concrete results we are forced to resort to various idealizations and approximations and the most basic of these is to separate chemistry from statistical mechanics.

Quantum field theory

½

A fundamental tool in the application of quantum mechanics to chemistry is the Born–Oppenheimer approximation that the masses of the nuclei are infinitely heavier than the mass of the electron. This allows the separation of A) the electronic energy levels for which quantum mechanics is responsible for chemical binding into molecules from B) the interactions of the molecules which are responsible for the bulk phases of matter. The Coulomb interactions of electrons and nuclei which cause the formation of molecules the electrons must be treated quantum mechanically, and the fact that the kinetic and potential energy do not commute must be taken into account. However, when we consider interactions between molecules with some effective potential, the quantum effects of the kinetic energy can usually be ignored and the problem becomes one of classical statistical mechanics. However, the price which must be paid for this reduction to classical mechanics is that the potential energy may no longer be considered to be the sum of pair potentials. For example a charged ion can induce a dipole moment in a neutral atom. For such systems three and higher body potentials are needed to describe the fact that molecules are not elementary but are composite. Furthermore, even though the Coulomb potential is spherically symmetric the effective molecular potentials very often have a complicated angular dependence of which the tetrahedral bonds of the carbon atom are probably the most familiar. The study of bulk properties which depend on quantum mechanics such as superconductivity and the (fractional) quantum Hall effect are traditionally considered as part of condensed matter physics and not as part of statistical mechanics. This is in part because these inherently quantum mechanical phenomena are often treated at absolute zero. The physics that involves both quantum mechanics and phase transitions at nonzero temperatures is still poorly understood. For example there is as yet no generally accepted theory of the computation of the superconducting phase transition temperature of high Tc materials. The reduction of quantum mechanics with Coulomb interactions to effective classical interactions between molecules is beyond the scope of this book. See [3] for further references. In fact, even the rough qualitative picture presented above is not established with any mathematical rigor. Nevertheless we will in this book make use of this qualitative intuition, and much of our considerations will be for classical systems. This will be seen to be completely adequate for dealing with the bulk phenomena of freezing, ferromagnetism and critical phenomena.

1.4

Quantum field theory

Statistical mechanics is commonly associated with condensed matter physics. However, perhaps of equal importance is the understanding which has developed over the past 40 years that statistical mechanics and in particular classical statistical mechanics, is exceptionally closely related to Euclidean quantum field theory. The relation between these two apparently disconnected subjects is seen from the path integral formulation of quantum field theory where the expectation value of operators A is given as 1 A = [dφ]AeS/¯h with Z = [dφ]eS/¯h (1.72) Z

¾¼

Basic principles

where the integral [dφ] is a functional integral over some appropriate space and S (which is called the action) is some functional of the fields φ. These formulas for Z and A are identical with the formulas for the partition function and for averages in the canonical ensemble and thus if we identity h ¯ in quantum field theory with the temperature T of statistical mechanics we see that the two subjects have a total formal similarity. In this interpretation quantum fluctuations caused by h ¯ are analogous to temperature fluctuations caused by T. There are of course differences between the two subjects. First of all the functional integral [dφ] needs precise definition before calculations can be done. One such definition is to replace continuum space by a lattice of points and to start with finite volumes and take the thermodynamic limit. A second difference is in the choice of the action S as opposed to the classical potential U (r). Actions S tend to consist of a kinetic energy part and a single site potential energy while classical potentials tend to have a complicated two-body behavior. Nevertheless there are sufficient similarities that lattice gauge theories can be regarded as isomorphic to certain classical spin systems in statistical mechanics and the best known of the solvable lattice statistical models, the Ising model, has a precise quantum field theory analogue which will be discussed later in this book.

References [1] A.B. Pippard, The Elements of Classical Thermodynamics, Cambridge Univ. Press (1961). [2] D. Ruelle, Statistical Mechanics (Benjamin, New York 1969) chap. 6. [3] L.D. Landau and E.M. Lifshitz, Statistical Physics, third edition part 1 (Pergamon 1993). See especially p. 258, chap. XIII and chap XIV.

2 Reductionism, phenomena and models In chapter 1 we presented the fundamental philosophy of statistical mechanics. In this chapter we present the experimental phenomena for which we will use statistical mechanics to study in the remainder of this book. In order to do this we need to begin with a review of the concept of reductionism. We then present the types of phenomena which will be studied and conclude with a discussion of the theoretical models used to describe these experimental phenomena.

2.1

Reductionism

The methods used in science to study systems depend crucially on their size. The smallest length scales are studied by physics; larger scales are studied by chemistry; still larger scales are studied by biology; larger still is the scale of economics and the largest scale of all is the the universe which is the province of astrophysics and cosmology. Examples of this hierarchy are shown in Table 2.1. Table 2.1 Levels of reductionism with their degrees of freedom and the discipline which studies them.

Level Universe Societies Living organisms Cells Viruses Polymers Molecules Atoms Nuclei Particle physics Quantum chromodynamics String theory

Degrees of freedom Stars Stock prices Humans Amoeba, blood cells Polio, tobacco mosaic Hemoglobin, cholesterol Water, carbon dioxide Hydrogen, helium, argon Uranium and plutonium nuclei Protons, neutrons, electrons Quarks, gluons Strings, branes

Discipline Cosmology Economics Biology Biology Biochemistry Chemistry Chemistry Physics Physics Physics Physics Physics

Statistical mechanics can be used at any of these levels of description and indeed statistical methods and concepts are used from string theory to economics. But in this book we confine our attention to the level of atoms and molecules. Therefore we will consider nature to be described by

Reductionism

¾¿

The Practical Person’s Theory of Everything The nonrelativistic quantum mechanics of nucleons and electrons interacting by means of Coulomb and magnetic interactions This theory of everything excludes such things as nuclear fission, black holes and quantum gravity but in principle it is sufficiently general and powerful to explain most physical phenomena from the scale of atoms through the scale of societies. For the practical person this is more than enough. Unfortunately our ability to use this theory for detailed calculations is extremely limited. Even the computation of the energy levels of helium must be done numerically, and computations of the ground state energy of the electron gas at finite density have only been carried to a few orders before unpleasant logarithms arise. Consequently what appear to be the simplest problems in atomic and condensed matter physics at zero temperature already involve serious computational complexity. How then is it possible to actually use the formalism of quantum statistical mechanics at positive temperatures and pressures which was presented theoretically in the preceding chapter? The answer to this is an integral part of the philosophy of reductionism. Each scale is governed by its own set of laws, and reductionism is the attempt to derive the laws of the larger system from the laws of the smaller system. Thus string theory tries to explain quantum chromodynamics; quantum chromodynamics tries to explain nuclear physics; atoms are built from nuclei and electrons; molecules are built from atoms and so on. The levels of description are characterized by what are referred to as degrees of freedom: quarks and gluons for quantum chromodynamics, protons and neutrons for nuclear physics, the elements of the periodic table for atomic physics on up to human beings in societies. These degrees of freedom interact with each other. The behavior of a large number of degrees of freedom in interaction with each other is what is studied by statistical mechanics. Therefore on the scale of atoms we will, for the purposes of statistical mechanics, not attempt to compute the properties of atoms from the theory of everything. This is left to the field of atomic physics. Similarly we will not attempt to compute the properties of molecules from the theory of everything. This is the province of chemistry. We will not attempt to compute the interaction potentials between these degrees of freedom. The laws of interaction will either be taken from experiment or, more commonly, simple force laws will be assumed which are hoped to incorporate the crucial features needed to explain the phenomena we are interested in. Furthermore, since atoms such as argon and molecules such as water are heavy when compared to electrons and nuclei, quantum mechanics in terms of both the symmetry of the wave functions and the non-commutativity of kinetic and potential energy can often be safely ignored. Thus for many purposes we will describe the statistical mechanics of atoms and molecules in terms of classical mechanics where we specify the position and momentum of the degrees of freedom and characterize the interactions between them by a (possibly many-body) potential U (r1 , . . . , rN ). This description is only expected to be valid in some restricted range of temperature and pressure, and some care must be taken not to extend the statistical mechanics to regions of tempera-

Reductionism, phenomena and models

ture and pressure where the underlying description of the degrees of freedom becomes invalid.

2.2

Phenomena

The phenomena of greatest interest in statistical mechanics are phase transitions and critical phenomena. We here survey some of these phenomena with an emphasis on their phase diagrams. 2.2.1

Monatomic insulators

The elements with the simplest experimentally observed phase diagram are the noble gases neon (Ne), argon (Ar), krypton (Kr) and xenon (Xe) in the region of pressure and temperature where they may be considered as monatomic insulators. This phase diagram is given schematically in Fig. 2.1 where we show the equation of state as a surface in the P T v space. In Fig. 2.2 we plot the projections of the phase boundaries in the P T and P v planes The lines indicate the phase boundary where there is a first order phase transition between two coexisting phases which are either solid/liquid or liquid/gas. The curve in the P v plane which separates the liquid and gas phases is known as the coexistence curve. The critical point (Pc , Tc , vc ) is at the end of the first order line in the P T plane which separates the liquid/gas phases. It is possible to go from the gas to the liquid phase in a continuous path which does not cross the first order line and thus the liquid and gas phases are sometimes collectively referred to as the fluid phase. The triple point (Pt , Tt ) is a point in the P T plane where three first order lines meet and where the solid, liquid and gas phases coexist with specific volumes vs , vl and vg . We give the measured critical point and triple point data for neon, argon, krypton and xenon in Table 2.2. The phase diagrams of these noble gases have been measured up to pressures of 6.0 GPa. These results are plotted in Fig. 2.3. On the scale of this figure the critical and triple point are not visible. The solid phase of all the noble gases is the fcc (face centered cubic) lattice. The definition of the fcc and all the other Bravais lattices is recalled in the appendix. Table 2.2 Critical and triple point data for neon, argon, krypton and xenon taken from [2]. The unit for the pressure is 1 bar= 105 N/m2 = 105 Pa, the unit for specific volume is cm3 /mole and we note that atmospheric pressure is 1.013 bar.

Ne Ar Kr Xe

Tc (K) 44.4 150.7 209.5 289.7

Pc (bar) 26.5 48.6 55.2 58.4

vc 41.8 74.9 91.3 118.0

Tt (K) 24.55 83.80 115.76 161.39

Pt (bar) 0.4332 0.6893 0.7298 0.8160

vg 4600 9867 12850 16050

vl 16.18 28.24 34.32 44.31

vs 14.06 24.6333 30.013 38.59

Phenomena f e

d ID

LIQU

LIQUID

D OLI

AL CRITIC POINT

LIQ U VA ID – PO R

c

E

E

b

R

LIN

PO VA

IPL

SO LID

AS

TR

G

SOLID

–

PRESSURE

S

–V AP OR

a

T4 Tc

T2

VO

LU

T3

RE

T1

ME

TU

RA

E MP

TE

LIQUID

CRITICAL POINT

LIQUID

SOLID SOLID

PRESSURE

PRESSURE

S-L

SOLID - LIQUID

Fig. 2.1 The phase diagram in P T v space of a system which has a critical point and a triple point following [1].

CRITICAL POINT

GAS L-V GAS

LIQUID VAPOR VAPOR

S-V VAPOR SOLID-VAPOR TEMPERATURE

VOLUME

Fig. 2.2 (a)The projection on the P T plane of the P T v phase surface of Fig. 2.1. (b) The projection on the P v plane of the P T v phase surface of Fig. 2.1 following [1]. The critical point is indicated by (Pc , Tc , vc ) and the triple point by (Pt , Tt , vs , vl vg ).

2.2.2

Diatomic insulators

Many familiar elements such as nitrogen, oxygen and the halogens – fluorine, chlorine, iodine and bromine – exist at room temperature and pressure not as single atoms but as diatomic molecules. These diatomic molecules each have two additional rotational degrees of freedom in their kinetic energy which in the ideal gas approximation leads to a specific heat at constant volume of 5kB /2 instead of the 3kB /2 which characterizes the monatomic ideal gases. The interactions between these diatomic molecules now depends on the orientation of the molecules relative to each other. Diatomic insulators have liquid–gas transitions and critical points just as do the monatomic insulators. We give the critical data for oxygen, nitrogen and the halogens in Table 2.3.

Reductionism, phenomena and models 6 RARE GAS MELTING CURVES 5 He

Ne Ar

P (GPa)

4

3

2 Kr 1

Xe Rn

0

200

0

400 T (K)

600

Fig. 2.3 Freezing/melting curves for the noble gases at high pressure following [3]. The unit of pressure is GPa= 109 N/m2 . Note that 1 bar= 10−4 GPa and atmospheric pressure is 1.013 × 10−4 GPa so that the critical and triple points schematically shown in Figs. 2.1 and 2.2 will not be visible when plotted on this high pressure scale.

The phase diagrams of diatomic molecules are more complicated than the noble gases and show great variety which indicates that there are other properties beyond their diatomic nature which are important. We illustrate this diversity by giving in Figs. 2.4 and 2.5 the phase diagrams for fluorine, oxygen and nitrogen. The properties of the different phases of O2 and N2 are given in Tables 2.4. Table 2.3 Critical point data for oxygen, nitrogen and the halogens taken from [3]. The unit for the pressure is 1 bar = 105 N/m2 = 105 Pa and we note that atmospheric pressure is 1.013 bar.

O2 N2 F2 Cl2 Br2 I2

Tc (K) 154.6 126.2 144.3 417 584 819

Pc (bar) 50.43 30.39 52.15 77.0 103 103

vc (cm3 /mole) 73.4 89.5 66.2 124.7 127 155

Phenomena

FLUORINE

P (GPa)

4

cm(4) (α)

sc(8) (β)

2

Liquid

0 0

300

200

100 T (K)

Fig. 2.4 The phase diagram of F2 following [3]. The number in parenthesis indicates the number of molecules per unit cell. The symbols indicate the space group symmetry in the notation of appendix A.

20 z

OXYGEN

NITROGEN

20

15 15

P (GPa)

P (GPa)

cm(2) (ε)

10

fco(4) (δ)

5

rh(1) (β)

rh(8) (ε)

10

Liquid

st(2) (γ)

5

cm(2) (α) 0

0

200

Liquid

hcp (b)

sc(8) (γ)

d¢

sc(8) (δ)

sc(4) (α) 400

T (K)

600

0

0

200

400

600

800

T (K)

Fig. 2.5 Phase diagrams for O2 and N2 following [3]. The number in parenthesis indicates the number of molecules per unit cell. The symbols indicate the space group symmetry in the notation of appendix A.

Reductionism, phenomena and models Table 2.4 Properties of the phases of oxygen and nitrogen shown in Fig. 2.5 from [3].

Phase α β γ δ δ Phase α β γ δ 2.2.3

Oxygen Properties Monoclinic, antiferromagnetic Rhombohedral, possible two-dimensional short range helicoidal order Paramagnetic Orthorhombic Existence unclear Monoclinic Nitrogen Properties Orientationally ordered Hex close packed, no orientational order Molecules in layers with common orientation Cubic Rhombohedral Liquid crystals

Another very important class of materials where the molecular degrees of freedom behave as hard rods are organic liquid crystals. These materials are very different from the diatomic molecules in that the rods can have an orientational ordering even though the center of mass is still in a liquid state. There are three types of phases observed. Nematic: The centers of mass have no order and form a liquid, while the rods have an orientational order. Smectic: The centers of mass are ordered in layers and the rods are oriented perpendicular to the layers. Cholesteric: The centers of mass are ordered in layers and the rods are ordered parallel to the layers where the direction of ordering smoothly rotates as one goes from one layer to the next. 2.2.4

Water

The most familiar of a polyatomic molecule is H2 O, water. The molecular structure of water is that the two hydrogen atoms bond to the oxygen atom with an angle of 105 degrees between the bonds. The triple and critical point data of water are given in Table 2.5. Water has the well-known but very atypical property that at atmospheric pressure it expands when it freezes. Furthermore there are many different phases of ice. The phase diagram is sketched in Fig. 2.6 in P T v space and in Fig. 2.7 in P T space. Table 2.5 Critical and triple point data for water. The unit for the pressure is 1 bar= 105 Newtons/m2 = 105 Pa and we note that atmospheric pressure is 1.013 bar.

H2 O

Tc (K) 647.30

Pc (bar) 2.2 × 103

Tt (K) 273.16

Pt (bar) 6.0 × 10−2

Phenomena

0 10 C °

.6°

81 °

L & IQU IC ID E VI I

60 °

40 °

20 0°

AN

000

15,

ICE

20,

00

10,0

LIQ U

ID

000

PRESSURE

0 ,00 2 20 m kg/c

D

VI

VI ICE

VII

AN D

–

–

° 40

ICE

VII

° 20

VI

15,

000

UID

00

LIQ

50

10,

000

50

00

50

V

kg

0 60

II

0

III

80

EC

&

0 90

VO

&

II

C

0

LU

ME

T “P RIP L LIQUID OI NT E & ICE I ” I 0°

10

00

III

IFI

I

I

SP

R

PO

D AN

V UID

0 70

VI

&

/cm2

11

cm 0 3 0 /kg

0°

–4

–2

LIQ

VA

0°

C

0 ° 1

.6 81

°

60 °

40

°

20

E UR AT ER P M TE

0°

Fig. 2.6 The phase diagram for water and ice following [1].

2.2.5

Metals

Metals are characterized by the property of high electrical conductivity and therefore, unlike the monatomic and diatomic insulators, the degrees of freedom in the metallic phase must consist of ions and electrons instead of neutral molecules. Thus the phase diagrams for metals should be described by a theory much closer to the underlying Coulomb Hamiltonian than potentials used to describe neutral molecules. We show the phase diagrams of copper (Cu), gold (Ag) and silver (Au) in Fig. 2.8 and potassium (K) and sodium (Na) in Fig. 2.9. They illustrate the general rule that the phase diagrams of most metals are simpler than most diatomic and polyatomic insulators. The critical point data of sodium and potassium are shown in Table 2.6. Table 2.6 Critical point data for sodium and potassium from [3]. The unit for the pressure is 1 bar= 105 Newtons/m2 = 105 Pa. The unit of specific volume is cm3 /mole.

Na K

2.2.6

Tc (K) 2485 2198

Pc (bar) 255 150

vc (cm3 /mole) 76.6

Helium

In the phase diagrams presented thus far the only property of the nucleus which played any role was the atomic number. However, in helium at low temperatures it makes a profound difference if the nucleus is He3 or He4 and this is one of the few elements where

¿¼

Reductionism, phenomena and models 8000 ice VI

6000 ice V liquid (water)

4000 ice II

ice III

p /atm

2000

218

critical point

ice I

2 liquid (water)

ice I

vapour (stream)

1 triple point 0.006 0 200

300 273.15 273.16 (T 3) (T f)

400 373.15 (T b)

500 T /K

600

700 647.30 (T c)

Fig. 2.7 Phase diagram in P T space for water and ice following [6]. The pressures are given in atmospheres, Tf , T3 and Tb are respectively the temperatures of freezing, the triple point and boiling point at the pressure of one atmosphere.

the nonrelativistic Coulomb interaction must be supplemented by quantum mechanical considerations of symmetry which distinguish He3 , which obeys Fermi statistics, from He4 , which obeys Bose statistics. The low temperature phase diagrams for these two systems are shown in Fig. 2.10. For He3 there is only one normal liquid phase in this temperature range whereas He4 has a second order phase transition between a normal liquid I and a superfluid phase II. At much lower temperatures He3 also shows a superfluid phase. The phase diagram at higher temperatures and pressures is shown in Fig. 2.11 where the difference between the two isotopes has become merely qualitative. The liquid/gas critical points of He3 and He4 are given in Table 2.7. Table 2.7 Critical point data He3 and He4 taken from [3]. The unit for the pressure is 1 bar= 105 N/m2 = 105 Pa and we note that atmospheric pressure is 1.013 bar.

He3 He4

2.2.7

Tc (K) 3.310 5.190

Pc (bar) 1.147 2.275

vc (cm3 /mole) 72.5 57.54

Magnetic transitions

There are two other types of very common phase transitions which need to be discussed: ferromagnetism and antiferromagnetism. For these phenomena spin degrees

Phenomena

¿½

7 COPPER GROUP 6 Cu

5

P (GPa)

Ag 4

Au

3

fcc

2 Liquid 1 0

0

500

1500

1000

2000

T(K)

Fig. 2.8 The phase diagrams of copper, gold and silver following [3]. 12 16

SODIUM POTASSIUM 10

fcc (II) 12

P (GPa)

P (GPa)

8

8 Liquid

bcc (I)

6 bcc 4

4

Liquid 2

0

hex(9) 0

200

400 T (K)

600

800

0

0

200

400 T (K)

600

800

Fig. 2.9 The phase diagrams of potassium and sodium following [3].

of freedom must be included in the Hamiltonian and these degrees of freedom will interact with an external magnetic field. Phase transitions can take place in the spin degrees of freedom which are separate and distinct from the solid, liquid, gas transitions previously discussed. Ferromagnetism The ferromagnetic phase is characterized by the alignment of the spin degrees of freedom below a temperature Tc called the Curie temperature. Elements which have ferromagnetic phase transitions are shown in Table 2.8. Magnetism is, of course, an extremely important phenomenon and there are a very large number of alloys and compounds which are ferromagnetic. In Table 2.8 we compare the temperature Tc with the crystal phase structure at atmospheric pressure. This table reveals that

¿¾

Reductionism, phenomena and models 5

25 HELIUM – 4

4

HELIUM – 3

20

hcp

hcp bcc

3

P (MPa)

P (MPa)

15 Liquid I

2

Liquid

10 Liquid II bcc

1 5

0

0

1

2

3

0

T (K)

2

0

4

6

T (K)

Fig. 2.10 The low temperature phase diagram of He3 and He4 following [3]. 1.0 HELIUM

0.8

0.6 P (GPa)

hcp fcc

0.4

He – 3 He – 4

0.2 Liquid

0

0

10

20 T (K)

30

Fig. 2.11 The high pressure phase diagram of He3 and He4 from [3].

the ferromagnetic transition occurs substantially below the temperature at which the material has solidified into a lattice. It is therefore often possible to consider the spin degrees of freedom separately from the translational degrees of freedom.

Models

¿¿

Table 2.8 Critical (Curie) temperatures of elements which show ferromagnetism at atmospheric pressure. The phases of the lattice and the temperatures of transition are shown for comparison. The Curie temperatures are from [4] and the remaining properties from [3].

Material Fe Co Ni Gd Dy

Tc (K) 1043 1388 627 293 85

Phase boundaries bcc R and has finite continuous derivatives of all orders. Then the Fourier transform fˆ(k) is an entire function of k and has the property that, for all N and |k|, |fˆ(k)|
R, D ik·r ˆ dD reik·r f (r). (3.253) f (k) = d re f (r) = |r|≤R

The boundedness of the integration region ensures that all derivatives of fˆ(k) exist for |k| < ∞ and thus fˆ(k) is an entire function of k. We next note that the existence of derivatives of all orders guarantees the existence of ∂N dD reik·r f (r) (3.254) ∂rα1 · · · ∂rαN |r| R the function F (r) is a function of a complex variable r which is analytic in a strip which includes the real r axis, then the Fourier transform will vanish exponentially as |k| → ∞ with k real. To illustrate this consider:

Appendix B: Fourier transforms

Example B2. Proof of (3.71) Let f (r) = (r2 + 1)−D .

(3.268)

Then fˆ(k) =

dre

ir·k

2

−D

(r + 1)

∞

= 0

=π

1/2

D−2 ) Γ( 2

rD−1 dr 2 (r + 1)D

π

dθ sinD−2 θeikr cos θ 0

D2 −1 ∞ rD/2 2 dr 2 J D (kr) r (r + 1)D 2 −1 0

(3.269)

where we have used (3.263). Using the Sonine-Gegenbauer integral [6, (51) on p.95] we find the desired result: D/2 π 1/2 Γ( D−1 k ir·k 2 −D 2 ) dre (r + 1) = KD/2 (k) > (3.270) D 2 Γ( 2 ) We note that k D/2 KD/2 (k) is bounded as k → 0 and that as k → ∞ k D/2 KD/2 (k) ∼ (π/2)1/2 k (D−1)/2 e−k .

(3.271)

References [1] D. Ruelle, Classical statistical mechanics of a system of particles, Helv. Phys. Acta 36 (1963) 183. [2] M.E. Fisher, Free energy of a macroscopic system, Arch. Rat. Mech. Anal. 17 (1964) 377–410. [3] M.E. Fisher and D. Ruelle, The stability of many-particle systems, J. Math. Phys. 7 (1966) 260–270. [4] D. Ruelle, Statistical Mechanics, (Benjamin 1969) chapter 3. [5] A. Lenard and S. Sherman, Stable Potentials II, Comm. Math. Phys. 17 (1970), 91–97. [6] A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions (McGraw-Hill New York 1953) vol. 2 [7] N. Angelescu, G. Nenciu and V. Protopopescu, On stable potentials, Comm. math. Phys. 22 (1971) 162–165. [8] D. Ruelle, Statistical mechanics of quantum systems of particles Helv. Phys. Acta 36 (1963) 789. [9] F.J. Dyson and A. Lenard, Stability of matter I, J. Math. Phys. 8 (1967) 423–434. [10] F.J. Dyson, Ground–state energy of a finite System of charged Particles, J. Math. Phys. 8 (1967) 1538–1545. [11] A. Lenard and F.J. Dyson, Stability of matter II, J. Math. Phys. 9 (1968) 698–711. [12] E. H. Lieb, The stability of matter, Rev. Mod. Phys. 48 (1976) 553–569. [13] E.H. Lieb, The N 5/3 law for bosons, Phys. Lett. 70A (1979) 71–73. [14] J.G. Conlon, E.H. Lieb and H.–T. Yau, The N 7/5 law for charged bosons, Comm. Math. Phys. 116 (1988) 417–448. [15] The stability of matter: from atoms to stars; selecta of Elliott H. Lieb ed. W. Thirring, (Springer-Verlag, Berlin and Heidelberg 1991). [16] L. Van Hove, Quelque propriétés génerales de l’intégral de configuration d’un system de particules avec interaction, Physica 15 (1949) 951–961. [17] R.P. Boas, Entire Functions (New York: Academic Press 1954). [18] J. Lebowitz and F. Dyson, Existence of thermodynamics for real matter with Coulomb forces, Phys. Rev. Letts. 22 (1969) 631–634. [19] C.N. Yang and T.D. Lee, Statistical theory of equations of state and phase transitions I. Theory of condensation, Phys. Rev. 87 (1952) 404–409. [20] M.E. Fisher and J.L. Lebowitz, Asymptotic free energy of a system with periodic boundary conditions, Comm. Math. Phys. 19 (1970) 251–272. [21] T.D. Lee and C.N.Yang, Statistical theory of equations of state and phase transitions II. Lattice gas and Ising model, Phys. Rev. 87 (1952) 410–419. [22] D. Ruelle, Superstable interactions in classical statistical mechanics, Comm. Math. Phys. 18 (1970) 127–159

References

½

[23] G.W. Milton and M.E. Fisher, Continuum fluids with a discontinuity in the pressure, J. Stat. Phys. 32 (1983) 413–438. [24] R.B. Griffiths and D. Ruelle, Strict convexity (“continuity”) of the pressure in lattice systems, Comm. Math. Phys. 23 (1971) 169–175. [25] M.E. Fisher, On the discontinuity of the pressure, Comm. Math. Phys. 26 (1972) 6–14. [26] M. Reed and B. Simon, Methods of Mathematical Physics II: Fourier analysis and self-adjointness (Academic Press 1975), p. 13.

4 Theorems on order In chapter 2 we presented several models which we want to use to describe freezing/melting transitions and critical phenomena in continuum systems and ferro- and antiferromagnetism in lattice magnetic systems. However, merely writing down a model does not demonstrate that it will actually exhibit the desired ordering, and in this chapter we turn our attention to a survey of the proofs of ordering for these potentials. A chronology of some ordering theorems is presented in Table 4.1. Studies of order date back to antiquity but in more modern times we may perhaps date the beginnings of the subject to the conjecture of Kepler [1] in 1611 that there is no denser packing of hard spheres than the face centered cubic lattice. Gauss proved [2] in 1831 that if we only consider periodic lattice packing that face centered cubic has maximum density but the demonstration that there is no nonperiodic structure that is denser than the fcc lattice was only given by Hales [12] in 2005. We will discuss these results in section 4.1. An obvious extension of the Kepler problem is to hard ellipsoids of revolution such as we discussed in connection with models of liquid crystals. In 2004 an example was discovered by Donev, Stillinger, Chaikin and Torquato [11] of a configuration of hard ellipsoids which, for all aspect ratios, is denser than the density of fcc close packed hard spheres. This is also presented in section 4.1. The modern studies of crystalline and magnetic order begin with the conjectures made on physical grounds by Peierls [13] and Landau [14] that in D = 1, 2 for T > 0 crystalline order does not occur and the proof by Peierls [3] in 1936 of the existence of spontaneous magnetization in the two-dimensional Ising model, The proof of this conjecture of nonexistence of crystalline order and the similar result of nonexistence of ferromagnetic and antiferromagnetic order in Heisenberg magnets was made in the mid 1960s and will be proven in detail in sections 4.2 and 4.3. The proof of existence of spontaneous magnetization in the Ising model was first given by Peierls [3] in 1936. However, we will not give this proof because the spontaneous magnetization for the Ising model on the square lattice will be computed exactly in chapter 12. The existence of order in Heisenberg models is substantially more difficult to prove. For the classical case a proof of the existence of ferromagnetic and antiferromagnetic order in D = 3 at T > 0 for isotropic nearest neighbor cubic lattices was first given in 1976 [7]. We present the proof in section 4.4. The extension to the quantum case has only been done for antiferromagnets. The case S ≥ 1 is done in [8] and S = 1/2 in [10]. At T = 0 antiferromagnetic order has been proven [9] in D = 2 for all S ≥ 1. We survey these proofs in sections 4.5 and 4.6.

Densest packing of hard spheres and ellipsoids

¿

It is obvious that Table 4.1 does not include answers to many questions which are raised in the previous chapter. This is not because we have only made a limited selection of results but because there are indeed a large number of unanswered questions. We explicitly formulate a few of these important “missing theorems” in section 4.7. Table 4.1 Chronology of selected theorems, conjectures and examples of order in D = 1, 2, 3.

Date 1611 1831 1936

Authors Kepler [1] Gauss [2] Peierls [3]

1940 1966

Fejes Tóth [4] Mermin, Wagner [5] Mermin [6] Froehlich, Simon, Spencer [7] Dyson, Lieb, Simon [8]

1968 1976

1978

1986

Neves, Perez [9]

1988

Kennedy, Lieb, Shastry [10] Donev, Stillinger, Chaikin, Torquato [11] Hales [12]

2004

2005

4.1

Theorem or conjecture Conjecture: fcc is densest structure for D = 3 Proof: fcc is densest lattice for D = 3 Proof: existence of ferromagnetic order in Ising model for D = 2 Proof: hexagonal is densest structure for D = 2 Proof: lack of order in D = 1, 2 Heisenberg ferro- and antiferromagets for T > 0 Proof: lack of crystalline order in D = 1, 2 for T > 0 Proof: existence of order in D ≥ 3 classical Heisenberg ferro- and antiferromagets for T > 0 for nearest neighbor isotropic cubic lattices Proof: existence of antiferromagnetic order in quantum Heisenberg antiferromagnets for S ≥ 1 at T > 0 for D = 3 nearest neighbor isotropic cubic lattices Proof: existence of antiferromagnetic order in the S ≥ 1 Heisenberg antiferromagnet at T = 0 for D = 2 nearest neighbor isotropic square lattices Proof: existence of antiferromagnetic order in the S = 1/2 Heisenberg antiferromagnet at T > 0 for D = 3 nearest neighbor anisotropic cubic lattices Example: unusually dense packings of ellipsoids

Proof: Kepler’s conjecture


One of the oldest problems of order in condensed matter physics is the question of the closest packed arrangement of spheres of diameter σ in D dimensions. In two dimensions this closest packed arrangement of hard discs is shown in Fig. 4.1. It is a translationally invariant lattice where each point has six nearest neighbors. This number of nearest neighbors is referred to by mathematicians as the kissing number. The fraction of space occupied by a configuration of spheres in dimension D is called the packing fraction. It is easy to compute √ that the packing fraction for the infinite hexagonal lattice in D = 2 of Fig. 4.1 is π/ 12 = 0.9069 · · ·. The space in which the discs are situated can be separated into cells which surround the centers of the discs

Theorems on order

which contain the points which are closest to the center of the disc. This decomposition of space into hexagonal cells is shown in Fig. 4.2. These cells are called Voronoi cells by mathematicians and Wigner–Seitz cell by physicists.

a

a c

b

b a

a

a c

c b a

a

Fig. 4.1 The hexagonal lattice of closest packed discs in D = 2.

Fig. 4.2 The Voronoi or Wigner–Seitz cells for closest packed hard discs.

A three-dimensional lattice may be constructed by stacking these hexagonal lattices on top of each other. In Fig. 4.1, where we denote the location of the centers of the discs of the first lattice as a, the maximum density is achieved if the centers of the spheres of second layer are at either the positions b or c. The third layer may also be added in two possible ways in a close packed fashion. Thus every sequence of the


letters a, b, c gives a lattice with the same density. The sequences of three positions · · · abcabcabc · · · or · · · acbacbacb · · ·

(4.1)

gives the face centered cubic lattice (fcc) and the sequence of two positions · · · ababab · · ·

(4.2)

gives the hex close packed lattice. The unit cells of these lattices are given in appendix A of chapter 2. Both of these lattices are translationally √ invariant, and each of them has 12 nearest neighbors and a packing fraction of π/ 18 = 0.7405 · · · . For the fcc lattice there is one site per unit cell which means that every lattice site is equivalent to every other lattice site. For the hcp lattice there are two sites per unit cell. It was proven by Gauss [2] that there is no translationally invariant lattice in D = 3 with a packing fraction greater than the fcc lattice. The question of the densest translationally invariant packing of spheres in D dimensions has been extensively investigated and many results are tabulated in the book by Conway and Sloane [15]. Of course, most packings of spheres will not be translationally invariant. However in D = 2 it was proven by Fejes Tóth [4] in 1940 that the hexagonal lattice is indeed the densest packed lattice. In D = 3 the conjecture of Kepler [1] of 1611 asserts that none of these non translationally invariant lattices is more densely packed than the fcc lattice. The proof of this was an outstandingly difficult problem for almost 400 years and was only proven in 2005 by Hales [12] in a series of papers of great complexity. The reason for this difficulty is that it is possible √ to find non-lattice packings which are locally more dense than the fcc density of π/ 18 and thus it is not obvious that a non-lattice packing must be less dense than a regular lattice packing. In fact for sufficiently large dimension D there are examples where non-lattice packings are more dense than regular lattice packings. Hales first showed that it was sufficient to check the density of a finite (but large) number of configurations and then the properties of these configurations were painstakingly computed on a case by case basis with the assistance of a computer. Because of the computer-assisted nature of the proof there was not total agreement in the mathematics community [16] as to whether the proof could be considered logically complete and the controversy stands as a landmark in the philosophy of mathematical proof. At high pressure, atoms of any material will be pressed together at high density and thus it may be expected that the repulsive core of the potential will play a dominant role in determining the crystal structure of the solid at high pressure. In chapter 2 we saw that many monotonic insulators and metals have fcc or hcp crystal structure at high pressure on the solid side of the melting curve. This is certainly consistent with a hard sphere model for the core of the potential of these systems. A particularly simple model for the liquid crystals discussed in chapter 2 is the hard ellipsoid of revolution with aspect ratio α = a/b shown in Fig. 4.3, and thus it is interesting to find the densest packing of such molecules. If we consider translationally invariant configurations with only one ellipsoid per unit cell then it can be shown, that for any aspect ratio, the maximum packing fraction is the same as the fcc lattice of hard spheres. However, if we allow several ellipsoids per

Theorems on order

unit cell with at least two inequivalent orientations it was found by Donev, Stillinger, Chaikin and Torquato [11] that more dense packings are possible. The densities √ they found depend on the aspect ratio α and are plotted in Fig. 4.4. For α ≥ 3 and √ α ≤ 1/ 3 the packing fraction is 0.770732 and the ellipsoids each have 14 nearest neighbors which is to be compared with the 12 nearest neighbors of the fcc lattice. However, to quote [11] “There is nothing to suggest that the crystal packing we have presented here is indeed the densest for any aspect ratio other than the trivial case of spheres.” This extension of Kepler’s conjecture remains very much an open question and much more work need to be done to understand the ordering properties of even this most simple model of a liquid crystal.

Fig. 4.3 An ellipsoid of revolution with aspect ratio α = a/b.

0.78 0.77

Packing fraction f

0.76 0.75 0.74

FCC

0.73 0.72 0.71 0.7 0.4

0.6

0.8

1

1.2 1.4 Aspect ratio a

1.6

1.8

2

Fig. 4.4 The packing fraction as a function of the aspect ratio for the packings of ellipsoids of Donev, Stillinger, Chaikin and Torquato taken from [11].

Lack of order in the isotropic Heisenberg model in D = 1, 2

4.2


The discussion of densest packings of section 4.1 are all geometric properties which have nothing to do with temperature. However, we are principally interested in the dependence of order on temperature, and the possibility that long range order exists for T > 0 depends very much on the dimension D of the system. We begin our considerations with the study of the isotropic quantum mechanical Heisenberg model of spin S introduced in chapter 2 which is defined by the Hamiltonian z H=− J(R − R )SR · SR − H SR (4.3) R. R

R

where the interaction constants are chosen without loss of generality to satisfy J(R) = J(−R) J(0) = 0.

(4.4)

The spin matrices at the site R are characterized by y y y x z z x z x , SR [SR ] = iSR δR,R , [SR , SR ] = iSR δR,R , [SR , SR ] = iSR δR,R

(4.5)

k† k = SR SR

(4.6)

S2 = S(S + 1).

(4.7)

and It is convenient to further introduce the notation −1/2 (SxR ± iSyR ) S± R = 2

(4.8)

which from (4.5) satisfy the commutation relations − z [S+ R , SR ] = SR δR,R ,

± [SzR , S± R ] = ±SR δR,R

(4.9)

For ease of notation we consider cubic lattices where R = N where N is a Ddimensional vector with D components and R is restricted to the box Ω, R ∈ Ω if 0 ≤ Nj ≤ Lj − 1,

(4.10)

and we impose periodic boundary conditions on (4.3) by setting Nj = Lj ≡ 0. The total number of sites is N = j Lj . The extension to other lattices is merely a matter of extending the notation to other Bravais lattices. The magnetization is defined as 1 z M z (H; N ) = SR (4.11) N R

where we recall that the thermal average is given by X = TrXe−βH /Tre−βH .

(4.12)

From (4.3) and (4.11) we see that M z (−H; N ) = −M z (H; N ). z

(4.13)

When the size of the system N is finite then M (H; N ) must be continuous (in fact analytic) at H = 0 and thus for finite N we must have M z (0; N ) = 0 in any dimension.

Theorems on order

However, we are interested in the thermodynamic limit N → ∞. In this limit continuity and analyticity at H = 0 are no longer guaranteed and thus we define the spontaneous magnetization (ferromagnetic order) as M z (0+ ) = lim

lim M z (H; N ).

H→0+ N →∞

(4.14)

When M z (0+ ) is positive we say that the magnet has ferromagnetic order. The existence or nonexistence of ferromagnetic order in the isotropic Heisenberg magnet depends on the dimension D. The first theorem we prove for this Heisenberg magnet is the theorem of Mermin and Wagner [5]. Theorem 4.1: Mermin and Wagner If we impose on the Heisenberg model of (4.3) the additional restriction R2 |J(R)| < ∞

(4.15)

R

then the magnetization is bounded above for sufficiently small fields H by in D = 1 const T −2/3 |H|1/3 |M z | < const T −1/2 | ln |H||−1/2 in D = 2.

(4.16)

Therefore, when (4.15) holds the isotropic Heisenberg magnet has no spontaneous magnetization. We will prove theorem 4.1 by use of the following inequality of Bogoliubov [17]. Lemma 4.1: Bogoliubov’s inequality For any operators A, C and H = H† 1 AA† + A† A[[C, H], C † ] ≥ kB T |[C, A]|2 . 2

(4.17)

To prove the inequality (4.17) we first define a quantity which has come to be called the Duhamel two-point function [8]. Definition: Duhamel two-point function The Duhamel two-point function (A, B) may be defined by (A, B) =

i|A|j∗ i|B|j

i,j

Wi − Wj Ej − Ei

(4.18)

where Ei are the eigenvalues of H, the sum is over all pairs of the eigenstates of H excluding pairs with equal energy, and Wj = e−βEj /Tre−βH .

(4.19)

We first prove that the Duhamel two-point function has the two properties needed for an inner product (A, B) = (B, A)∗

(4.20)


0 < (A, A).

(4.21)

Property (4.20) is obvious from the definition (4.18). To prove property (4.21) we note that β e−βEi − e−βEj = tanh (Ej − Ei ) (4.22) e−βEi + e−βEj 2 and tanh β2 (Ej − Ei ) ≤ β/2. (4.23) 0≤ Ej − Ei It therefore follows that 0≤ and since

Wi − Wj β ≤ (Wi + Wj ) Ej − Ei 2

i|A|j∗ i|A|jWi =

i,j

i|AA† |iWi = AA†

(4.24)

(4.25)

i

we have the two inequalities

0
0.

Hs →0+ N →∞

(4.64)

The proof that there is no antiferromagnetic order in dimension D = 1 and 2 is proven by first setting setting ˜ + (k), C=S

˜ − (−k − K) A=S

(4.65)

in the Bogoliubov inequality (4.17). The rest of the proof is identical with the proof given in the ferromagnetic case.

4.3

Lack of crystalline order in D = 1, 2

The principal model introduced in the previous chapter to study monotonic insulators is the classical N-body Hamiltonian 1 2 p + U (N ) (r1 , · · · rN ) 2m j=1 j N

H=

(4.66)

½

Theorems on order

where the potential U (N ) (r1 , · · · rN ) is the sum of two-body pair potentials U (N ) (r1 , · · · rN ) =

1 U (ri − rj ). 2

(4.67)

i=j

with U (r) = U (−r). The conditions on U (r) needed for the existence of the thermodynamic limit have been extensively discussed in chapter 3. The nonexistence of crystalline order for this classical system was investigated by Mermin [6] in 1968 by means of a classical analogue of the Bogoliubov inequality (4.17). The argument is more involved than what was done in the previous section for the Heisenberg magnet in two respects: 1) The restrictions on the potential are more subtle than (4.15) and 2) the characterization of crystallization is more involved than the definition of spontaneous magnetization (4.14). Restrictions on the potential We proved in chapter 3 that for the thermodynamic limit to exist it is sufficient that   C1 /|r|D+ for 0 ≤ |r| ≤ a1 for a1 < |r| < a2 U (r) ≥ −C2 (4.68)  −C3 /|r|D+ for |r| > a2 with the constants Cj > 0. However for the proof of nonexistence of crystalline order for D = 1, 2 of [6] to hold we need the further restrictions that: 1) U (r) is twice differentiable for r = 0. In particular, this excludes the case of any potential with a hard core. 2) The following bounds must hold with Cj > 0 1 ∂ D−1 ∂ r U (r) ≥ −C4 /rD+2+ as r → ∞ rD−1 ∂r ∂r ∂ ∂ U (r) − λr2 |∇2 U (r)| = U (r) − λr3−D | rD−1 U (r)| ≥ C5 /rD+ ∂r ∂r for some λ > 0 as r → 0

∇2 U (r) =

(4.69)

(4.70)

D 2 2 where ∇2 = i=1 ∂ /∂ri and in the last term before the inequality sign we have 2 specialized ∇ to the spherically symmetric case where U (r) = U (|r|). The LennardJones potential U (r) = (σ/r)n − A(σ/r)m for n > m > D (4.71) and the oscillatory potential U (r) = cos K · r/rn for n > D + 2

(4.72)

satisfy (4.69) and (4.70) while a potential which diverges at r → 0 as U (r) = eB/r does not.

(4.73)


½

Definition of crystalline order The physical concept of crystalline order is that the atoms are situated on the vertices of a periodic lattice such as fcc, bcc, or cubic in D = 3, or square or hexagonal in D = 2. The location and orientation of the crystal are obviously arbitrary, and in order to give a precise mathematical formulation of crystalline order we need to localize the position and orientation of the supposed lattice. We will do this by enclosing the N particles in a box with impenatrable walls of a shape consistent with the shape of the supposed crystalline order and take the limit as the size of the box goes to infinity. To be specific we concentrate on D = 2 and assume that the crystal has a Bravais lattice n1 a1 + n2 a2 (with ni integers) specified by the two vectors a1 and a2 in a box commensurate with the Bravais lattice specified by the points r = x1 N1 a1 + x2 N2 a2 ∈ Ω with 0 ≤ x1 , x2 ≤ 1

(4.74)

where Nj are integers, the volume of Ω in D = 2 is V = N1 N2 |a1 ×a2 | and the total number of particles N is N = nN1 N2 where n is the number of particles per unit cell. The density at the point r when the N particles are at rj is ρˆ(r) =

N 1 δ(r − rj ) V j=1

(4.75)

and thus the integrated average density in D = 2 is ρ0 = n/|a1 ×a2 |. The Fourier transform of ρˆ(r) is ρ˜(k) =

1 V

dre−ik·r ρˆ(r) =

Ω

N 1 −ik·rj e . V j=1

(4.76)

Thus we define the thermal average of the k component of the density as ρ(k) = ˜ ρ(k)

(4.77)

where, for the classical system (4.66), the thermal average of a quantity f is defined as (N ) 1 dr1 · · · drN e−βU (r1 ,···rN ) f (r1 · · · rN ) (4.78) f = QN Ω

with QN =

dr1 · · · drN e−βU

(N )

(r1 ,···rN )

.

(4.79)

Ω

The vector K is said to be in the reciprocal lattice of the Bravais lattice specified by ai if (4.80) K = K1 b1 + K2 b2 with bj · ak = 2πδj,k . If there is no crystalline order the density will be uniform in the thermodynamic limit, but if there is a periodic crystalline order there will be at least one vector in the reciprocal lattice for which ρ(k) does not vanish. Thus we have the following criteria:

½

Theorems on order

Crystalline order exists if and only if A) B)

lim ρ(k) = 0 k is not a reciprocal lattice vector

N →∞

lim ρ(K) = 0 for at least one nonzero reciprocal lattice vector

N →∞

(4.81)

where limN →∞ denotes the thermodynamic limit. We may now prove the theorem of Mermin [6]: Theorem 4.2 Crystalline order in the sense of (4.81) does not exist for the classical Hamiltonian (4.66) in D = 1, 2 when the potential U (r) satisfies restrictions 1 and 2 given above. We prove the theorem by using a Classical analogue of the Bogoliubov inequality Let ψ(r) be a differentiable function and φ(r) be a twice differentiable function which vanishes when r is on the surface of the box. Then we have kB T | j φ(rj )∇j ψ(rj )|2 2 | (4.82) ψ(rj )| ≥ 1 2 i,j |φ(ri ) − φ(rj )|2 ∇2j U (ri − rj ) + kB T j |∇j φ(rj )|2 j where we use the notation ∇j = ∂/∂rj . To prove (4.82) we start from the obvious inequality valid for any scalar function A(r1 , · · · , rN ) and vector function B(r1 , · · · , rN ) 2

B−

AA∗ B ≥0 |A|2

from which it follows that |A|2 ≥

|A∗ B|2 . |B|2

(4.83)

(4.84)

Choose A and B to be A=

N

ψ(rj )

j=1 (N )

eβU B=− β

(4.85)

N N ∂ ∂U (N ) 1 ∂φ(rj ) −βU (N ) φ(rj ) (4.86) (φ(rj )e )= − ∂rj ∂rj β ∂rj j=1 j=1

and note that using the first expression for B in (4.86) we have, for any differentiable function X(r1 , · · · , rN ), BX = −

1 βQ

dr1 · · · rN Ω

N j=1

X

N ∂ −βU (N ) 1 ∂X e φ(rj ) = φ(rj ) ∂rj β j=1 ∂rj

(4.87)

where to obtain the last line we have integrated by parts and used the fact that φ(rj ) vanishes on the boundary of the box. From (4.87) we find

Lack of crystalline order in D = 1, 2 N 1 ∂ψ ∗ (rj ) BA = φ(rj ) β j=1 ∂rj ∗

and |B|2 = B · B∗ =

N 1 ∂ φ(rj ) · B∗ . β j=1 ∂rj

½

(4.88)

(4.89)

Then, using the second expression for B of (4.86) in (4.89), we find N 1 ∂ ∂ (N ) |B| = φ(ri )φ∗ (rj ) · U β i,j=1 ∂ri ∂rj 2

∗

N ∂φ (rj ) ∂U (N ) 1 2 ∗ 1 φ(rj ) · − ∇j φ (rj ) . + β j=1 ∂rj ∂rj β

(4.90)

The second term in (4.90) can be written as N N ∗ 1 1 ∂φ(rj ) 2 ∂ −βU (N ) ∂φ (rj ) − 2 = 2 dr1 · · · drN φ(rj ) · e | | (4.91) β Q Ω ∂rj ∂rj β j=1 rj j=1 where the last term is obtained by integrating by parts and noting that φ(rj ) vanishes on the boundary. Thus |B|2 =

N N 1 ∂ ∂ (N ) 1 ∂φ(rj ) 2 φ(ri )φ∗ (rj ) · U + 2 | | β i,j=1 ∂ri ∂rj β j=1 ∂rj

(4.92)

and therefore using N i.j=1

φ(ri )φ∗ (rj )

N ∂ ∂ ∂ (N ) 1 ∂ · U = |φ(ri ) − φ(rj )|2 · U (ri − rj ) (4.93) ∂ri ∂rj 2 i.j=1 ∂ri ∂ri i=j

we obtain the desired inequality (4.82) by using (4.88), (4.92) and (4.93) in (4.84). Proof of theorem 4.2 We now prove theorem 4.2 by letting K be the vector in the reciprocal lattice where the density ρ(K) is assumed not to vanish in the thermodynamic limit and specialize (4.82) by setting φ(r) = sin k · r and ψ(r) = e−i(k+K)·r (4.94) where k=n ˜ 1 b1 /N1 + n ˜ 2 b2 /N2

(4.95)

˜ i ≤ Ni /2 for Ni even and (Ni − 1)/2 ≤ n ˜ i ≤ (Ni − 1)/2 for Ni odd. with Ni /2 < n Thus we have |

N j=1

ψ(rj )|2 = V 2 ˜ ρ(k + K)˜ ρ(−k − K)

(4.96)

½

Theorems on order

|

N

φ(rj )∇j ψ(rj )|2 =

j=1

V2 (k + K)2 |˜ ρ(K) − ρ˜(K + 2k)|2 4

(4.97)

and

N 1 |φ(ri ) − φ(rj )|2 ∇2i U (ri − rj ) + kB T |∇j φ(rj )|2 2 i,j=1 j i=j

=

N N 1 (sin k · ri − sin k · rj )∇2j U (rj − rk ) + kB T k2 cos2 k · rj ) 2 i,j=1 j=1 j=k

≤

N 1 (ri − rj )2 ∇i U 2 (ri − rj ) + N kB T 2 i,j=1

(4.98)

i=j

where to obtain the last line of (4.98) we have used (sin k · rj − sin k · rk )2 ≤ k2 (rj − rk )2

and

cos2 k · r ≤ 1

(4.99)

and hence using (4.96)–(4.98) in (4.82) we obtain ρ(K) − ρ˜(K + 2k)2 (k + K)2 /4 kB T ˜ 1 ˜ ρ(K + k)˜ ρ(−K − k) ≥ 2 . (4.100) N k (kB T + (1/2N ) i,j (ri − rj )2 ∇2i U (ri − rj )) Thus far the size of the system specified by Nk and the number of particles N have been finite. To complete the proof we need to take the thermodynamic limit Nk , N → ∞. We first prove that N 1 (ri − rj )2 ∇2i U (ri − rj ) < ∞. N1 ,N2 ,N →∞ 2N i,j=1

lim

(4.101)

i=j

It is here that we will make use of the restrictions on the potential (4.69) and (4.70). To prove (4.101) we define a new N -body potential (N )

Uλ

(r1 , · · · , rN ) = U (N ) (r1 , · · · , rN ) − λδU (N ) (r1 , · · · , rN ) 1 Uλ (ri − rj ) 2 i,j N

=

(4.102)

i=j

where Uλ (r) = U (r) − λr2 ∇2 U (r). (N ) Uλ

(4.103)

For we have the configurational contribution to the free energy per particle fλ for the finite system


−βfλ (N ) =

1 ln N

from which we find − and

dr1 · · · drN e−β(U

(N )

−λδU (N ) )

,

(4.104)

Ω

∂fλ (N ) = δU (N ) λ ≥ 0 ∂λ

(4.105)

∂ δU (N ) λ = β(δU (N ) − δU (N ) 2λ λ ≥ 0 ∂λ (N )

where · · ·λ denotes a thermal average with respect to Uλ we find 1 λ f0 (N ) − fλ (N ) = δU (N ) µ dµ N 0 ≥ λδU (N ) 0 =

½

(4.106)

. From (4.105) and (4.106)

N λ (ri − rj )2 |∇2j U (ri − rj )| ≥ 0. 2N i,j=1

(4.107)

i=j

Therefore to prove (4.101) it is sufficient to prove the existence of the two thermodynamic limits f0 = lim f0 (N ) < ∞ and fλ = lim fλ (N ) < ∞. N →∞

N →∞

(4.108)

The proofs of chapter 3 demonstrate that the free energies per site f0 and fλ exist if the two body potentials U (r) and Uλ (r) each satisfy (4.68). Thus, noting that the restriction (4.68) for Uλ (r) is the restrictions (4.69) and (4.70), the proof of (4.101) follows. Thus using (4.107) in (4.100) we obtain 1 ρ(K) − ρ˜(K + 2k)2 (k + K)2 /4 kB T ˜ ˜ ρ(K + k)˜ ρ(−K − k) ≥ . N k2 (kB T + (f0 − fλ )/λ)

(4.109)

To complete the proof we define g(q) = e−αq

2

with α > 0

(4.110)

multiply both sides of (4.109) by g(k + K), divide by the volume V and sum over the vectors k = 0 specified by (4.95). Each term in the sum on the right is positive and thus the right-hand side of the inequality is made smaller by restricting the sum of the vectors k = 0 such that 2|k| is less that the length K0 of the shortest reciprocal lattice vector. For the vectors in this restricted sum the quantity ˆ ρ(k + K) vanishes in the thermodynamic limit by criterion A in the assumption of crystalline order. Therefore we find from (4.109) that 1 kB T K2 g(K0 /2)ρ(K)2 1 ˜ ρ(q)˜ ρ(−q) ≥ g(q) V q N 16[kB T + (f0 − fλ )/λ] V We now note that

0=k |k| 0.

r→∞ L→∞

This is equivalent to n lim

L→∞

j=1

1 j SR N

(4.120)

2 = 0

(4.121)

R∈Ω

which is written in terms of the Fourier transform (4.32) as lim

L→∞

1 ˜2 S (0) = 0. N

(4.122)

The proof of the existence of long range ferromagnetic order (4.122) for the classical Heisenberg magnet (4.115) depends both on the properties of J(R) and on the lattice on which R takes its values. However, the known cases where existence can be proven do not seem to exhaust the cases where it would seem that order should exist. Consequently we will first present the mechanism used for the proofs of long range order and after that we demonstrate that under certain restrictive assumptions the bounds used in the mechanism can be proven to hold. 4.4.1

The mechanism for ferromagnetic order

Long range order (4.122) will follow from the bound 1 | J(R − R )(hj (k · R) − hj (k · R ) · (SR − SR )|2 2 R,R ∈Ω |hj (k · R) − hj (k · R )|2 J(R − R ) ≤ kB T

(4.123)

R,R ∈Ω

where hj (k · R)|m = δjm N −1/2 eik·R are n component vectors We use the bound (4.123) by fixing k = 0 noting that |hj (k · R) − hj (k · R )|2 J(R − R ) = (1 − eik·R )J(R) = 2Ek R,R ∈Ω

and

R∈Ω

(4.124)

½½¾

Theorems on order

1 2 =

J(R − R )(hj (k · R) − hj (k · R )) · (SR − SR )

R,R ∈Ω

J(R − R )(hj (k · R) − hj (k · R )) · SR

R,R ∈Ω

=

eik·R j ˜j (k) SR J(R )(1 − eik·R ) = 2Ek S 1/2 N R∈Ω R ∈Ω

(4.125)

where we have defined Ek =

1 1 (1 − eik·R )J(R) = (1 − cos k · R)J(R) 2 2 R∈Ω

(4.126)

R∈Ω

˜ and used the definition (4.32) of the Fourier transform S(k). Thus we see that (4.123) specializes when k = 0 to ˜j (k)S ˜j (−k) ≤ 2kB T Ek 0 ≤ 4Ek2 S

(4.127)

or, in other words, for k = 0 ˜ j (k)S ˜j (−k) ≤ 0 ≤ S

kB T . 2Ek

(4.128)

We now note that the normalization condition (4.116) on SR may be written in ˜ terms of the Fourier transform S(k) as 1 1 ˜ ˜ 1 = S2R = S2R = S(k) · S(−k). (4.129) N N R∈Ω

k

Then if we separate the sum over k in (4.129) into the terms k = 0 and k = 0 and use the bound (4.128) for k = 0 we obtain 1 ˜ 2 1 ˜ ˜ 1 = S(0) + S(k)S(−k) N N k=0

kB T n 1 1 1 ˜ 2 + . ≤ S(0) N 2 N Ek

(4.130)

k=0

We now take the thermodynamic limit where the sum in (4.130) may be replaced by an integral. Thus 3 1 ˜ 2 d k kB T n S(0) + (2π)−3 . (4.131) 1 ≤ lim L→∞ N 2 Ek But the integral in (4.131) is finite and thus for T sufficiently small we must have lim

L→∞

1 ˜ 2 S(0) > 0 N

which is the condition (4.122) for long range order.

(4.132)

Existence of ferromagnetic and antiferromagnetic order

½½¿

In the special case of nearest neighbor isotropic interactions on the cubic lattice where Ek = J(3 − cos k1 − cos k2 − cos k3 ) the integral in (4.131) has been analytically evaluated by Watson [19] as

π

π

π

1 = 3 − cos k − cos k2 − cos k3 1 −π −π −π √ √ √ √ √ 2 √ (18 + 12 2 − 10 3 − 7 6)[ K((2 − 3)( 3 − 2))]2 = 0.505462 · · ·(4.133) π where K(k) is the complete elliptic integral of the first kind 1 dx . (4.134) K(k) = 2 2 2 1/2 0 [(1 − x )(1 − k x )] −3

(2π)

dk1

dk2

dk3

Thus we find that long range order exists when 3.95678 . (4.135) kB T /J < n In Table 4.2 we compare this bound with numerical values determined from high temperature series expansions (see chapter 9). Table 4.2 Comparison of the lower bound on kB Tc /J computed from (4.135) and the values obtained from high temperature series expansion of chapter 9 for the Ising model n = 1 and the classical Heisenberg model n = 3.

n 1 3 4.4.2

Lower bound 3.95678 1.31893

Tc from series expansion 4.5108 1.44 ± 0.02

Proof of the bound (4.123)

It remains to formulate the set of restrictions on the interaction strengths J(R) for which the bound (4.123) can be proven true. At the very least the desired inequality (4.128) puts the necessary condition on J(R) that 1 Ek = (1 − cos k · R)J(R) > 0 for all k = 0. (4.136) 2 R∈Ω

The most natural restriction on the J(R) is the following Conjecture The necessary positivity condition (4.136) is also sufficient for the bound (4.123) to hold. There is no general proof of this conjecture and we will content ourselves with the special case proven by Fr¨ ohlich, Simon and Spencer [7] in 1976 of nearest neighbors on the cubic lattice: J(R) = J > 0 for R = (±1, 0, 0), (0, ±1, 0), (0, 0, ±1) 0 otherwise

(4.137)

½½

Theorems on order

This case is extremely restrictive and it is impossible to believe that it exhausts the sets of J(R) for which long range ferromagnetic order exists for the classical Heisenberg model. For example, it seems entirely reasonable to expect that the addition of positive bonds between any two sites can never decrease the long range order and thus, we would expect that long range order would exist on all cubic lattices with J(R) ≥ 0. This would be the case if, for the completely general lattice H=−

1 2

J(R, R )SR · SR ,

(4.138)

R,R ∈Ω

we could prove that ∂ SR1 · SR2 ∂J(R3 , R4 ) = (SR1 · SR2 )(SR3 · SR4 ) − SR1 · SR2 SR3 · SR4 ≥ 0.

kB T

(4.139)

The inequality (4.139) has been proven true for J(R, R ) ≥ 0 for cases that SR has one component (the Ising model) by Griffiths [20] and for the two-component case (the rotator model) by Ginibre [21], but for three or more spin components, even though some inequalities for correlation functions have been obtained [22], the inequalities (4.139) have never been proven. On the other hand no counter example to (4.139) has been found for the case of three or more spin components and thus there is nothing to suggest that (4.137) is the only case with nonnegative interactions J(R) where long range order exists. What seems to be lacking is the mathematical machinery to confirm our physical intuition. We will here prove that the bound (4.123) holds for isotropic interactions on the nearest neighbor cubic lattice (4.137). We begin with the following elementary lemma: Lemma 4.2 For any positive F (x) and G(x) and real h we have ∞ 2 0≤ dxdyF (x)e−βJ(x−y+h) /4 G(y)| ≤ ||F ||β ||G||β

(4.140)

−∞

where

∞

dxdyF (x)e−βJ(x−y)

2

/4

−∞

F (x) ≡ ||F ||2β .

(4.141)

To prove this lemma we define the Fourier and inverse Fourier transform ∞ ∞ 1 ikx ˜ F (k) = dxe F (x), F (x) = dke−ikx F˜ (x) (4.142) 2π −∞ −∞ and write ∞

∗

dxdyF (x)e −∞

−βJ(x−y+h)2 /4

∞

G(y) = −∞

−k2 /βJ ihk ˜ dk F˜ ∗ (k)G(k)e e

(4.143)


where we have used ∞ 2 dxe−βJ(x−h) eikx = eikh −∞

∞

dxe−βJx

2

/4 ikx

e

= eikh e−k

2

/βJ

½½

(4.144)

−∞

which holds for βJ > 0. Furthermore if we note that |eihk | = 1 we find from (4.143) that ∞ ∞ 2 −βJ(x−y+h)2 /4 dxdyF (x)e G(y) ≤ dxdyF (x)e−βJ(x−y) /4 G(y). (4.145) −∞

−∞

Then noting that the positivity of (4.141) is sufficient to prove the Schwarz inequality ∞ 2 dxdyF (x)e−βJ(x−y) /4 G(y) ≤ ||F ||β ||G||β . (4.146) −∞

the inequality (4.140) follows. We now follow [7, 18, 23] and use lemma 4.2 to prove a bound called Gaussian domination. Theorem 4.4. Gaussian domination If we define Z(hj (k · R))   β = dSR δ(S2R − 1)exp − J(R − R )[SR − SR − hj (k · R) + hj (k · R )]2  4 R

R,R

(4.147) and if J(R) satisfies the restriction (4.137) of nearest neighbors on the cubic lattice with periodic boundary conditions then for all hj (k · R) Z(hj (k · R)) ≤ Z(0).

(4.148)

To prove this theorem we consider splitting the lattice into two pieces of equal size which transform into each other by reflection through a plane perpendicular to the 1 (x) axis which bisects the bonds between R1 = 0, 1 and the bonds R1 = L, −L + 1 ≡ L + 1 and split the Hamiltonian into three parts: those terms where −L + 1 ≤ R1 , R1 ≤ 0, those terms where 1 ≤ R1 , R1 ≤ L and those terms R1 = 0, R1 = 1 and R1 = L, R = −L + 1. We introduce the notation dµR = dSR δ(S2R − 1) and write (4.147) as

(4.149)

½½

Theorems on order

Z(hj (k · R)) = dµ(0,R2 ,R3 ) dµ(1,R2 ,R3 ) dµ(L,R2 ,R3 ) dµ(−L+1,R2 ,R3 ) R2 ,R3

Z− [S(0,R2 ,R3 ) , S(−L,R2 ,R3 ) ] βJ exp − [S(0,R2 ,R3 ) − S(1,R2 ,R3 ) − hj (k · (0, R2 , R3 ) + hj (k · (1, R2 , R3 )]2 4 βJ [S(L,R2 ,R3 ) − S(−L+1,R2 ,R3 ) − hj (k · (L, R2 , R3 )) + hj (k · (−L + 1, R2 , R3 ))]2 exp − 4 Z+ [S(1,R2 ,R3 ) , S(L,R2 ,R3 ) ] (4.150) where Z− [S(0,R2 ,R3 ) , S(−L,R2 ,R3 ) ] = 3 βJ 2 [SR − SR+δi − h(k · R) + h(k · (R + δi ))] dµR exp − 4 −L+1≤R ≤−1 i=1 1 −L2 ≤R2 ≤L2 ,−L3 ≤R3 ≤L3

(4.151) and Z+ is given by (4.151) with 1 ≤ R1 ≤ L − 1. We now use the inequality (4.140) j j j j repeatedly with x, y the pairs of variables S0,R , S0,R and SL,R , S−L+1,R 2 ,R3 2 ,R3 2 ,R3 2 ,R3 and thus we find   dµ(0,R2 ,R3 ) d¯ µ(0,R2 ,R3 ) d¯ µ(−L+1,R2 ,R3 ) dµ(−L+1,R2 ,R3 ) Z(hj (k · R)) = R2 ,R3

βJ 2 [S(0,R2 ,R3 ) − S(1,R2 ,R3 ) ] Z− [S(0,R2 ,R3 ) , S(−L+1,R2 ,R3 ) ] exp − 4

1/2 βJ 2 ¯ ¯ ¯ [S(L,R2 ,R3 ) − S(0,R2 ,R3 ) ] Z+ [S(0,R2 ,R3 ) , S(−L+1,R2 ,R3 ) ] exp − 4   d¯ µ(1,R2 ,R3 ) dµ(1,R2 ,R3 ) dµ(L,R2 ,R3 ) d¯ µ(L,R2 ,R3 ) R2 ,R3

βJ ¯ 2 ¯ ¯ [S(1,R2 ,R3 ) − S(1,R2 ,R3 ) ] Z− [S(1,R2 ,R3 ) , S(L,R2 ,R3 ) ] exp − 4

1/2 βJ ¯ (L,R ,R ) ]2 Z+ [S(1,R ,R ) , S(L,R ,R ) ] (4.152) [S(L,R2 ,R3 ) − S exp − 2 3 2 3 2 3 4 ¯ R both indicate dummy variables of integration. It is easily recognized where SR and S that the two factors in (4.152) are equal and are each equal to the original expression for Z(hj (k · R)) with hj (k · (0, R1 , R2 )) − hj (k · (1, R1 , R2 )) and hj (k · (L, R1 , R2 )) − hj (k · (−L + 1, R1 , R2 )) set equal to zero.


½½

We now repeat this process for all other planes bisecting the lattice in the x, y and z directions. This sets all hj (k · R)) = 0 and thus the bound of Gaussian domination (4.148) is established. To complete the deduction of (4.123) from (4.148), we see from the definition (4.147) of Z(hj (k · R)) that   β J(R − R )(hj (k · R) − hj (k · R ))2  Z(hj (k · R)) = exp − 4 R,R   β dSR δ(S2R − 1) exp − J(R − R )(SR − SR )2  × 4 R R,R   β J(R − R )(SR − SR ) · (hj (k · R) − hj (k · R )) × exp  2 R,R

(4.153) and therefore noting that Z(0) is the partition function of the system we find from (4.148) and (4.153) that   β exp  J(R − R )(SR − SR · (hj (k · R) − hj (k · R )) 2 R,R   β J(R − R )(hj (k · R) − hj (k · R ))2  . ≤ exp  (4.154) 4 R,R

Finally we replace hj (k · R) by λhj (k · R) and expand (4.153) in λ. The term proportional to λ on the left-hand side vanishes because of translational invariance. Thus by equating the terms of λ2 we obtain  2 β 2 /2 J(R − R )(SR − SR ) · (hj (k · R) − hj (k · R )) R,R

≤β

J(R − R )(hj (k · R) − hj (k · R ))2

(4.155)

R,R

which holds when J(r) is given by (4.137). In the proof of (4.155) we have assumed that hj (k · R) is real but the final inequality extends to the case where hj (k · R) is given by (4.137). This establishes (4.123) and thus the proof is complete that long range order exists at sufficiently low temperature for the classical Heisenberg magnet on the nearest neighbor cubic lattice. 4.4.3

Antiferromagnetism

For the classical Heisenberg magnet on the nearest neighbor isotropic cubic lattice, if we make the transformation

½½

Theorems on order

SR = (−1)R1 +R2 +R3 SR

(4.156)

we send the spontaneous magnetization into the staggered magnetization and the ferromagnetic Hamiltonian transforms into the antiferromagnetic Hamiltonian H = −J

3

SR · SR+δi = J

R i=1

3

SR · SR+δi .

(4.157)

R i=1

Therefore if the ferromagnet has long range ferromagnetic order the antiferromagnet will also have antiferromagnetic order.

4.5

Existence of antiferromagnetic order in the quantum Heisenberg model for T > 0 and D = 3

The physics of the quantum Heisenberg magnet is substantially more involved than the classical case. For example we saw above that for the classical case the mapping (4.156) sends the ferromagnet into the antiferromagnet so the physics of these systems is essentially the same. However, in the quantum case the ferromagnet and the antiferromagnet are essentially different.This is easily seen by considering the very simple case of two spins of S = 1/2 with the interaction H = −JS1 · S2 . Explicitly written out as a 4 × 4 matrix this Hamiltonian is   1 0 0 0 J  0 −1 2 0  . H =−  4  0 2 −1 0  0 0 0 1

(4.158)

(4.159)

This matrix has a triply degenerate eigenvalue of −J/4 and a nondegenerate eigenvalue of +3J/4. Therefore the ground state for the ferromagnet J > 0 is triply degenerate while the ground state for the antiferromagnet J < 0 is nondegenerate. The existence of order is much more difficult to prove in the quantum mechanical spin S Heisenberg model than for the classical case. For the quantum ferromagnet even nearest neighbor interactions on the cubic lattice where H = −J

3

SR · SR+δ1

(4.160)

R i=1

with J > 0 and SR obeys the commutation relations (4.5), there is no proof of long range order for any spin S < ∞. For the quantum antiferromagnet on the nearest neighbor cubic lattice 3 H=J SR · SR+δ1 (4.161) R i=1

with J > 0, Dyson, Lieb and Simon [8] proved in 1978 that for the case S ≥ 1 long range antiferromagnetic order exists for some T > 0. The extension to S = 1/2 was

Existence of antiferromagnetic order in the quantum Heisenberg model for T > 0 and D = 3

½½

made by Kennedy, Lieb and Shastry [10] in 1988 who showed for a spatially anisotropic lattice J(1, 0, 0) = J(0, 1, 0), and J(0, 0, 1) = rJ(1, 0, 0) (4.162) that long range antiferromagnetic order exists in the ground state for 0.16 ≤ r ≤ 1

(4.163)

(and it is stated that the proof can be extended to some T > 0.) The proof of [8] will only here be briefly discussed. The proof first obtains a bound on the Duhamel two-point function, defined here as 1 (A, B) = Tr e−xβH Ae−(1−x)βH B /Tre−βH , (4.164) 0

of

3

˜j (k), S ˜j (−k)) ≤ (S

j=1

where in dimension D Ek = D +

D

3 2Ek

cos ki .

(4.165)

(4.166)

i=1

˜j (−k)) a bound is obtained on the thermal average ˜j (k), S From this bound on (S j j ˜ (k)S ˜ (−k) and then the mechanism of 4.4.1 is used to produce a lower bound TL S on the critical temperature Tc below which long range order occurs. This bound is obtained as the solution of S(S + 1) = (|EG |/2)(2π)−D dD k(Ek /Ek )1/2 coth[βL (2|EG |Ek Ek /3D)1/2 ] |ki |≤π

(4.167) where Ek = D −

D

cos ki

(4.168)

i=1

and EG is the ground state energy per site of the antiferromagnetic chain. The right-hand side of (4.167) is maximum at TL = 0 (βL = ∞). Therefore in order for antiferromagnetic order to exist the spin must satisfy the lower bound S(S + 1) ≥ (|EG |/2)1/2 ID where −D

ID = (2π)

|ki |≤π

dD k(Ek /Ek )1/2

(4.169)

(4.170)

which has been numerically evaluated in D = 3 as I3 = 1.157. Thus antiferromagnetic order will exist at finite temperature in D = 3 if S(S + 1) ≥ (3|EG |/2)1/2 1.157.

(4.171)

½¾¼

Theorems on order

It remains to find the ground state energy EG . In [8] a bound is given (first proven by Anderson in 1951 [24]) that in D dimensions |EG | < DS(S + 1/2) and thus in three dimensions antiferromagnetism exists for 1/2 3S(S + 1/2) S(S + 1) ≥ 1.157 2

(4.172)

(4.173)

which is satisfied for all S ≥ 1. Furthermore it was shown in [10] that the bounds can be improved to include the case S = 1/2 as well. Therefore it has been proven that, in D = 3 (and for all greater dimensions as well), for all spins S there is antiferromagnetic order in the quantum Heisenberg antiferromagnet in the nearest neighbor isotropic cubic lattice.

4.6

Existence of antiferromagnetic order in the quantum Heisenberg model for T = 0 and D = 2

We proved in section 4.2 that in two dimensions there is no long range order in either the spin S ferromagnet or antiferromagnet for T > 0. However, it is possible that there is long range order in the ground state even though order is impossible for T > 0. For the nearest neighbor Heisenberg antiferromagnet on the square lattice antiferromagnetic order has indeed been proven to exist for S ≥ 1 by Neves and Perez [9] by a method which extends the work of [8]. The case of D = 2 and S = 1/2 remains an open question but numerical evidence [25] suggests that antiferromagnetism does exist in the ground state.

4.7

Missing theorems

A comparison of the many phase diagrams in chapter 2 which exhibit crystalline, ferromagnetic, or antiferromagnetic order with the few actual proofs of order for specific models given in this chapter demonstrates that we do not have proofs of the existence of order for many, if not most, cases of physical interest. Some of this lack of knowledge is presented in Table 4.3 where we list some of the “theorems” which we would like to prove (or disprove). One of the glaring weaknesses in the proofs of order of sections 4.3–4.6 is that they crucially rely on the reflection properties of the lattice models. These restrictions were so severe that very few short range interactions are covered by the proofs and, for example, there is not even a proof of order for nearest neighbors on the bcc lattice. This restriction of the method to lattice systems makes it impossible to extend the proof of order in the one component classical magnet (Ising model) to the liquid–gas transition, crystallization or the existence of the solid–liquid–gas triple points of a genuine continuum fluid. In addition the restrictions on the proof of order in the classical Heisenberg magnet discussed in section 4.4 seem very artificial, and for the quantum Heisenberg ferromagnet, even though [8] conjectures very plausible bounds which lead to bounds on critical temperatures for Heisenberg ferromagnets, none of these conjectures have been proven.

Missing theorems

½¾½

Orientational order for liquid crystals has been studied for fluids that are confined to move on lattices [26] but no rigorous theorems on liquid crystal ordering for particles moving in the continuum seem to exist. It is very clear that the existence of order is a delicate and subtle phenomenon. This need for more exact knowledge of the general phenomena must be kept in mind in reading all the subsequent chapters on computations for specific models such as the high temperature series expansions for the Heisenberg model of chapter 9, which do indeed predict ferromagnetic order for the S = 1/2 model in three dimensions. Table 4.3 Missing “theorems”

1 2 3 4 5 6 7 8

Existence of crystalline order for D = 3 (Non)existence of crystalline order for hard discs in D = 2 Existence of the liquid–gas transition in continuum fluids Existence of the solid–liquid–gas triple point Existence of order in the classical Heisenberg model for general lattices Existence of ferromagnetic order in the quantum Heisenberg magnet for D = 3 Existence of antiferromagnetic order in the S = 1/2 Heisenberg magnet for D = 2 at T = 0 How does the spatial and orientational order of hard ellipsoids depend on D and on the aspect ratio α?

References [1] J. Kepler, De Niue Sexangula, in the Gesammelte Werke, Band IV, Kleinere Schriften 1602–1611 eds. M. Caspar and F. Hammer. C.H. Beck’sche Verlagsbuchhandlung (Munich 1901) 264–280. [2] C.F. Gauss,Untersuchen u ¨ ber die Eigenschaften der positiven tern¨ aren quadratischen Formen von Ludwig August Seeber, Göttische gelehrte Anzeigen, July 9, 1831. Reprinted in Werke, Vol. 2 K¨ oniglichen Gesellschaft der Wissenschaften zu G¨ ottingen, 1863, 188–196. [3] R. Peierls, On Ising’s model of ferromagnetism. Proc. Camb. Phil. Soc. 32 (1936) 477–481. [4] L. Fejes Tóth, Uber einen geometrischen Satz, Math. Z. 46 (1940) 79–83. [5] N.D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- or two- dimensional isotropic Heisenberg models, Phys. Rev. Letts. 17 (1966) 1133–1136. [6] N.D. Mermin, Crystalline order in two dimensions, Phys. Rev. 176 (1968) 250– 254. [7] J. Fr¨ ohlich, B. Simon and T. Spencer, Infrared bounds, phase transitions and continuous symmetry breaking, Commun. Math. Phys. 50 (1976) 79–85. [8] F.J. Dyson, E.H. Lieb and B. Simon, Phase transitions in quantum spin systems with isotropic and nonisotropic interactions, J. Stat. Phys. 18 (1978) 335–383. [9] E.J. Neves and J. F. Perez, Long range order in the ground state of two-dimensional magnets, Phys. Letts. A114 (1986) 331–333. [10] T. Kennedy, E.H. Lieb and B.S. Shastry, Existence of Néel order in some spin–1/2 Heisenberg antiferromagnets, J. Stat. Phys. 53 (1988) 1019–1030. [11] A. Donev, F.H. Stillinger, P.M. Chaikin and S. Torquato, Unusually dense crystal packing of ellipsoids, Phys. Rev. Letts. 92 (2004) 255506–(1-4). [12] T.C. Hales, A proof of the Kepler conjecture, Annals of Math. 162 (2005) 1065– 1185; T.C. Hales with S.P. Fergeson, The Kepler conjecture, Discrete and Computational Geometry 36 (2006) 1–269. [13] R.E. Peierls, On Ising’s model of ferromagnetism, Proc. Camb.. Phil. Soc. 32 (1936) 477–481. [14] L.D. Landau, Phys. Z. Sowjet. 11 (1937) 26. [15] J. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups (Springer Verlag 1999). [16] Nature, Vol. 424 (3 July 2003). [17] N.N. Bogoliubov, Physik. Abhandl. Sowjectunion 6 (1962) 1.113 and 229. [18] J. Fr¨ ohlich, R. Israel, E.H. Lieb and B. Simon, Phase transitions and reflection positivity I. General theory and long range lattice models, Comm. Math. Phys. 62 (1978) 1–34.

References

½¾¿

[19] G.N. Watson, Three triple integrals, Quart. J. Math. 10 (1939) 266–276. [20] R.B. Griffiths, Correlations in Ising ferromagnets, J. Math. Phys. 8 (1967) 478– 483. [21] J. Ginibre, General formulation of Griffiths’ inequalities, Comm.Math. Phys. 16 (1970) 310–328. [22] F. Dunlop, Correlation inequalities for multicomponent rotators, Comm. Math. Phys. 49 (1976) 247–256. [23] J. Fr¨ ohlich and T. Spencer, Phase transitions in statistical mechanics and quantum field theory, Cargese Lectures (1976). [24] P.W. Anderson, Limits on the energy of the antiferromagnetic ground state, Phys. Rev. 83 (1951) 1260. [25] Shoudan Liang, Existence of Néel order at T = 0 in the spin-1/2 antiferromagnetic Heisenberg model on a square lattice, Phys. Rev. 42 (1990) 6555–6560. [26] O.J. Heilmann and E.H. Lieb, Lattice models for liquid crystals, J. Stat. Phys. 20 (1979) 679–693.

5 Critical phenomena and scaling theory A dominant feature of the phase diagrams of chapter 2 is that there are many different type of ordering which occur and most materials have several different crystalline phases. The study of these different phases must depend on the details of the interaction potentials. But all of the elements whose phase diagrams are presented in chapter 2 share one common feature in that they all have a gas and liquid phase separated by a first order line which terminates in a critical point. Furthermore the discussion of the relation of the lattice gas model to the Ising model given in chapter 2 gives a mapping between certain models of ferromagnets and the liquid–gas transition and critical point. We will devote this chapter to the discussion of these phenomena. When we approach the critical point we find from experiment that thermodynamic properties such as the compressibility and the specific heat become infinite. This is taken to indicate that there are fluctuations in the system which extend over lengths which are much larger than the atomic size length scale and that at least some of the physics is insensitive to many of the details of the potential. There is thus the suggestion that there may be some properties of critical points which may in fact be independent of the material and might be predicted exactly by one of the simple models introduced in chapter 2. This is the origin of the concept of universality. We know much more about critical phenomena than we do of crystalline ordering. We saw in chapter 4 that we do not have a proof that crystalline order exists for any potential, much less do we have a model for which freezing transitions can be analytically studied in microscopic detail. It is therefore of enormous benefit for the study of critical phenomena that in two dimensions there is a special case of the latticegas model introduced in chapter 2 which has a critical point and a coexistence curve that can be solved exactly, because it is identical to the Ising model at zero magnetic field. The solvability of the two-dimensional Ising model in zero magnetic field is our most important tool in the study of critical phenomena. The calculations of exact properties of the Ising model will be presented in great detail in chapters 10–12 and from that study there emerges a very detailed microscopic picture of behavior near a critical point. From these results it is then possible to abstract a phenomenological picture which can be applied to other more realistic models such as the Ising model in three dimensions which can be hoped to be a realization of the hopes for universal properties of real three-dimensional fluids. The picture which emerges from this is referred to as scaling theory.

Thermodynamic critical exponents and inequalities for Ising-like systems

½

Scaling theory emerged in the 1960s from the independent work of several authors. Some of the principle papers are given in Table 5.1. It is one of the cornerstones of modern statistical mechanics and can be explained without reference to the Ising model. In that sense, scaling theory is a general principle. This is how it was developed, and this is how it will be presented here. Nevertheless it must be kept in mind that none of the considerations of the scaling theory of critical phenomena are rigorous in the sense of the proofs given in the last chapter. Scaling theory can perhaps best be described as a set of reasonable assumptions which have great plausibility because they are all proven to hold in the two-dimensional Ising model. In section 5.1 we begin the discussion of scaling theory by presenting the concept of critical exponents for Ising-like systems and proving that they are constrained by certain thermodynamic inequalities. In section 5.2 we present scaling theory for Ising systems, which, among other things, predicts equalities between the critical exponents. In section 5.3 we will extend our considerations from Ising models to Heisenberg models and Lennard-Jones fluids. In section 5.4 we introduce the concept of universality and conclude in section 5.5 with a discussion of missing theorems. Table 5.1 Chronology of the development of the scaling theory of critical phenomena.

Date 1959 1963 1963 1964 1964 1965 1965 1966 1967 1967

5.1

Reference Fisher [1] Rushbrooke [2] Essam, Fisher [3] Widom [4] Fisher [5] Griffiths [6], [7] Rushbrooke [8] Kadanoff [9] Fisher [10] Kadanoff et al. [11]

Development Critical exponent Critical exponent Critical exponent Scaling theory Scaling theory Critical exponent Critical exponent Scaling theory Scaling theory Scaling theory

equality inequality equality

inequalities inequality


We begin with the features of scaling theory which are obtained from the two-dimensional Ising model with nearest neighbor interactions E=−

Lv Lh

{E h σj,k σj,k+1 + E v σj,k σj+1,k + Hσj,k }

(5.1)

j=1 k=1

with σj,k = ±1. The most basic feature of the scaling theory of the critical point as seen in the Ising model (5.1) is that in fact there is a unique isolated point T = Tc , H = 0, M = 0 in the ferromagnet or T = Tc , P = Pc , v = vc in the continuum fluid at which the thermodynamic properties have singularities. This is already a nontrivial assumption,

½

Critical phenomena and scaling theory

because, for the inhomogeneous Ising model where interaction constants E h and E v are allowed to depend on the lattice position and vary randomly about some central value with some small standard deviation, it is known that the magnetization and susceptibility have singularities at different temperatures. (This will be discussed further in chapter 10.) Such random impurities simulate an impure (dirty) crystal, and thus it may be expected that for real materials the assumption that there is a single isolated Tc may fail. For the Ising ferromagnet the behavior of the thermodynamic functions is conventionally parameterized as shown in Table 5.2. Table 5.2 Critical exponent parameterization of the thermodynamic functions for the Ising– like magnets.

Property specific heat spontaneous magnetization susceptibility at H = 0 magnetization at Tc

Critical form cH ∼ Ac |T − Tc |−α cH ∼ Ac |T − Tc |−α M ∼ AM |T − Tc |β χ ∼ Aχ |T − Tc |−γ χ ∼ Aχ |T − Tc |−γ M ∼ AH H 1/δ

T → Tc + T → Tc − T → Tc − T → Tc + T → Tc − H → 0+

There are many comments which must be made about Table 5.2. The notation for the exponents is standard in the literature and originates in [5]. The exponents α and γ refer to T > Tc and α , γ and β are for T < Tc (and it is hoped that the exponent β will not be confused with β = 1/kB T ). There is no canonical notation for the amplitudes. We also note that it is a pure assumption that the singular parts of the thermodynamic functions are parameterized by pure algebraic powers instead of more complicated forms such as |T − Tc |−α lnp |T − Tc | or |T − Tc |−α lnp |T − Tc | ln ln |T − Tc |.

(5.2)

Indeed, the specific heat of the two-dimensional Ising model does have a logarithmic instead of an algebraic singularity. This logarithmic singularity is often “interpreted” as a limiting case of α = 0, and δ = 1 is often interpreted as ln H. But the most important point to be discussed about Table 5.2, for which there is no universal agreement in the literature, is the meaning of the symbol ∼. For the solvable case of the two-dimensional Ising model the free energy at H = 0, both above and below Tc , has the form F (T ) = F1 (T ) + F2 (T )(T − Tc )2 ln |T − Tc |

(5.3)

where F1 (T ) and F2 (T ) are analytic at T = Tc . Similarly the magnetization has an isolated singularity at Tc and in many (if not most) papers it is tacitly assumed that the meaning of the critical exponent forms is that there is an isolated algebraic or logarithmic singularity at Tc . However, we will see in chapter 12 that for the two-dimensional Ising model the possibility exists that the singularity in the magnetic susceptibility at


½

Tc is not isolated but may be embedded in a natural boundary. Therefore any conclusions drawn from a tacit assumption that the singularities are isolated must be treated with caution. These critical exponents are not independent but satisfy inequalities which follow from thermodynamics. Some of the most important of these thermodynamic inequalities are given in Table 5.3 Table 5.3 Thermodynamic inequalities for critical exponents in dimension D.

Date 1963 1965 1965 1969

Reference Rushbrooke [2] Griffiths [6] Rushbrooke [8] Griffiths [7]

Inequality α + 2β + γ ≥ 2 α + β(1 + δ) ≥ 2 γ (δ + 1) ≥ (2 − α )(δ − 1) γ ≥ β(δ − 1) D(δ − 1)/(δ + 1) ≥ (2 − η)

Gunton, Buckingham [12] Fisher [13]

As an example of these inequalities, we prove the inequality of Rushbrooke [2]. To do this we start from the magnetic analogue of the relation between cp and cv for a fluid 2 ∂M χ(cH − cM ) = T (5.4) ∂T H where the isothermal susceptibility χ is 2 ∂ F ∂M =− , χ= ∂H T ∂H 2 T

(5.5)

the two specific heats are cH = −T

∂2F ∂T 2

,

cM = −T

H

∂ 2 F˜ ∂T 2

(5.6) M

and we use the notation that F (T, H) denotes the free energy with T, H the independent variables and F˜ (T, M ) denotes the free energy with T, M the independent variables. There are now several cases. First consider lim cM /cH = R < 1.

T →Tc

(5.7)

Then using the critical exponent forms of Table 5.2 in (5.4) we find that as T → Tc − Aχ Ac (1 − R)|T − Tc |−γ

−α

(2 ' ∼ T AM β|T − Tc |β−1

(5.8)

and thus by matching the dependence on T − Tc on both sides of the equation we have the equality

½


α + 2β + γ = 2.

(5.9)

On the other hand, if R = 1 we expand with x > 0 cM /cH ∼ 1 + r|T − Tc |x

(5.10)

α + 2β + γ = 2 + x > 2.

(5.11)

and we find

For the case of the Ising model where the specific heat has a logarithm but the spontaneous magnetization and susceptibility are pure power laws we write at T → Tc cH ∼ Ac ln |T − Tc | + BH ,

and cM ∼ Ac ln |T − Tc | + BM

(5.12)

where BH and BM may be different and we then find from (5.4) that (2 ' Aχ |T − Tc |−γ (BH − BM ) = T AM β|T − Tc |β−1

(5.13)

from which we find γ = 2(1 − β)

(5.14)

BH − BM = Tc (AM β)2 /Aχ .

(5.15)

and

The remaining inequalities in Table 5.3 are proven in a similar fashion from convexity properties of thermodynamics in [6–8, 12, 13]. When scaling theory holds, all of these thermodynamic inequalities hold as equalities.

5.2

Scaling theory for Ising-like systems

Scaling theory extends the discussion of the description of critical phenomena from thermodynamic critical exponents to the position-dependent correlation functions. We begin by discussing the spin correlation function σ0 σR at zero magnetic field and consider the behavior for fixed T as |R| → ∞. For the Ising model in two dimensions we learn from the exact computations of chapters 11 and 12 that at H = 0 there are three separate cases which are summarized in Table 5.4 Table 5.4 Definition of the correlation exponents at H = 0 for Ising-like systems.

anomalous dimension correlation length

T = Tc T > Tc

correlation length

T < Tc

σ0 σR ∼ CTc /RD−2+η σ0 σR ∼ C> e−R/ξ> (T ) /Rp ξ> (T ) ∼ Aξ |T − Tc |−ν σ0 σR ∼ M 2 + C< e−R/ξ< (T ) /Rp ξ> (T ) ∼ Aξ |T − Tc |−ν


5.2.1

½

Scaling for H = 0

We begin by considering three separate cases of the R → ∞ behavior of the spin correlation function σ0 σR . Case 1. Asymptotic behavior of σ0 σR at T = Tc and H = 0 On a lattice away from Tc the correlations tend to have the symmetry of the lattice; square, cubic, etc. But it is found in the two-dimensional Ising model on the isotropic square lattice with nearest neighbor interaction energies E v = E h that as T → Tc for H = 0 the leading behavior of the correlations becomes rotationally invariant. More generally for the anisotropic lattice E v = E h at T = Tc the correlation has the leading behavior of CTc σ0 σRy ,Rx = 1/4 [1 + O(R−2 )] (5.16) R where (5.17) R2 = (sx Rx )2 + (sy Ry )2 where the correction term O(R−2 ) depends on sx Rx /sy Ry as well as on R. From this model result we formulate a general form for the leading asymptotic behavior of the correlations at T = Tc and H = 0 in dimension D of σ0 σR ∼

CTc . RD−2+η

(5.18)

The exponent η is known as the anomalous dimension. Case 2. Asymptotic behavior of σ0 σR at T > Tc and H = 0 When T > Tc and H = 0 the Ising spin–spin correlation function decays exponentially rapidly as R → ∞ and for T fixed we have σ0 σR =

C> (T ) −R/ξ> (T ) e [1 + O(R−1 )] R1/2

(5.19)

where as T → Tc + ξ> (T ) ∼ Aξ |T − Tc |−1

and C> (T ) ∼ c> |T − Tc |−1/4 .

(5.20)

The quantity ξ> (T ) is called the correlation length. From this model result we formulate a general form for the leading asymptotic behavior of the correlations at T > Tc in dimension D of C> (T ) −R/ξ> (T ) e (5.21) σ0 σR ∼ Rp where (5.22) ξ> (T ) ∼ Aξ |T − Tc |−ν .

½¿¼


Case 3. Asymptotic behavior of σ0 σR at T < Tc and H = 0 When T < Tc and H = 0 the Ising spin correlation σ0 σR approaches the square of the spontaneous magnetization, the approach to this limit is exponential and, for T fixed, we have σ0 σR = M 2 {1 +

C< (T ) −R/ξ< (T ) e [1 + O(R−1 )]} R2

(5.23)

where as T → Tc − ξ< (T ) ∼ Aξ |T − Tc |−1

and C< (T ) ∼ c< |T − Tc |−2 .

(5.24)

From this model result we formulate a general form for the leading asymptotic behavior of the correlation function for T < Tc in any dimension D of σ0 σR ∼ M 2 {1 +

C< (T ) −R/ξ< (T ) e } Rp

(5.25)

where as T → Tc − ξ< (T ) ∼ Aξ |T − Tc |−ν

and C< (T ) ∼ c< ξ< (T )p .

(5.26)

A most important feature of these three cases of the R → ∞ behavior of the correlation σ0 σR is that they are not uniform, in the sense that if T is set equal to Tc , neither the T > Tc form (5.19) nor the T < Tc form (5.23) reduces to the T = Tc result (5.16). Scaling theory attempts to unite these three cases together by defining the scaling limit and the scaling function. Scaling limit for H = 0 The scaling limit is defined at H = 0 as the limit T → Tc and R → ∞ with

and

lim R/ξ< (T ) = lim Aξ−1 R|T − Tc |ν = r for T < Tc

(5.27)

ν lim R/ξ> (T ) = lim A−1 ξ R|T − Tc | = r for T > Tc

(5.28)

T →Tc − R→∞

T →Tc + R→∞

T →Tc − R→∞

T →Tc + R→∞

fixed. Scaling function for T < Tc and H = 0 The scaling two-point function G− (r) is defined for T < Tc from the correlation function σ0 σR as G− (r) = lim M (T )−2 σ0 σR (5.29) T →Tc − scaling

where M (T ) is the spontaneous magnetization which is parameterized as T → Tc − by Table 5.2. From (5.25) and (5.26) we must have G− (r) ∼ 1 +

c< e−r rp

for r 1.

(5.30)


½¿½

There is no a priori reason why the large R behavior of < σ0 σR > at T = Tc must agree with the small r behavior of G− (r). Scaling theory makes the assumption that these two behaviors do in fact agree. Thus scaling theory assumes that G− (r) →

const as r → 0. rD−2+η

(5.31)

We then make contact with the large R behavior of σ0 σR at T = Tc by extending (5.29) to σ0 σR ∼ M (T )2 G− (R|T − Tc |ν ). (5.32) Then using the T = Tc form (5.18) on the left-hand side and Table 5.2 for M (T ) and (5.31) for the small r behavior of G− (r) on the right-hand side we find const const |T − Tc |2β ∼ . RD−2+η (R|T − Tc |ν )D−2+η

(5.33)

Because the left-hand side is independent of T − Tc the dependence of T − Tc on the right-hand side must cancel. Thus we find the exponent relation 2β = ν (D − 2 + η)

(5.34)

which we will see in chapter 12 holds analytically for the two-dimensional Ising model. Scaling function for T > Tc and H = 0 The scaling two-point function G+ (r) is defined for T > Tc from the correlation function σ0 σR as G+ (r) = lim const|T − Tc |−ν(D−2+η) σ0 σR T →Tc + scaling

(5.35)

where const is some normalizing constant, and scaling theory assumes that G+ (r) is finite and nonvanishing. To compare this with the T = Tc form (5.18) of σ0 σR we extend (5.35) to σ0 σR ∼ const |T − Tc |ν(D−2+η) G+ (R|T − Tc |ν )

(5.36)

and we see that in order to regain (5.18) we need G+ (r) ∼

const as r → 0. rD−2+η

(5.37)

Scaling functions for n-point correlationss We now may extend our definitions of scaling functions from the two to the n-spin correlations. For T < Tc we define G− (r1 , · · · , rn−1 ) = lim M (T )−n σ0 σR1 · · · σRn−1 (n)

scaling

(5.38)

and similarly for T > Tc we define G+ (r1 , · · · , rn−1 ) = lim |T − Tc |−nν(D−2+η)/2 σ0 σR1 · · · σRn−1 (n)

scaling

with the scaled lengths rj defined as in (5.27) and (5.28).

(5.39)

½¿¾


Relation of correlation to thermodynamic exponents We may now use these scaling functions to relate the thermodynamic exponents to the correlation exponents. For example consider the expansion of the magnetic susceptibility in terms of the two-point function ∂M (H) 1 χ(T ) = = {σ0 σR − M (T )2 }. (5.40) ∂H kB T H=0 R

As T → Tc − we use the scaling form (5.32) to write kB T χ(T ) ∼ M (T )2 {G− (R|T − Tc |ν ) − 1} R

∼ M (T )2 |T − Tc |−Dν

dD r{G− (r) − 1} −(Dν −2β) dD r{G− (r) − 1} ∼ const |T − Tc | ∼ const |T − Tc |−ν (2−η) dD r{G− (r) − 1}

(5.41)

where in the last line we used (5.34) and the constant depends on the details of the lattice interaction constants. Similarly for T → Tc + −ν(2−η) dD r G+ (r). kB T χ(T ) ∼ const |T − Tc | (5.42) Thus we find the prediction of scaling theory for the Ising model that the susceptibility exponents γ and γ satisfy γ = (2 − η)ν and γ = (2 − η)ν.

(5.43)

This relation was first obtained in 1959 by Fisher [1] for the two-dimensional Ising model. If we now use the exponent equality of Rushbrooke (5.9) with (5.34) and (5.43) we obtain the relation valid for T < Tc Dν = 2 − α . 5.2.2

(5.44)

Scaling for H = 0

There are no exact computations known for the Ising model with H = 0 but we may consider expanding the correlations for H = 0 in a power series in H/kB T by writing the interaction energy as E = E0 − H σR . (5.45) R

Thus

σ0 σR H = Z(H)−1

σ

σ0 σR e−E0 /kB T +H/kB T

R

σR

(5.46)


½¿¿

and by expanding the exponential we obtain σ0 σR H − M (H)2 =

∞ 1 (H/kB T )n σ0 σR σR1 · · · σRn cH=0 n! n=1

(5.47)

Rj

where the superscript c means that only the connected part of the correlation function is to be used. Then if we use the scaling form of the n-point function (5.38) and (5.39) we see that in order for a scaling function to exist we need the existence of the limit T → Tc and H → 0 of n 1 H M n+2 G− (|T − Tc |ν R, |T − Tc |ν R1 , · · · , |T − Tc |ν Rj ) n! kB T Rj n H const |T − Tc |β(2+n) |T − Tc |−Dν n dD r1 · · · drD ∼ n G− (r, r1 , · · · rn ) n! kB T (5.48) and hence we obtain for T < Tc the definition of the scaled magnetic field h = lim

H→0 T →Tc

H H = lim . ν (D+2−η)/2 H→0 |T − Tc |Dν −β |T − T | c T →Tc −

(5.49)

We have thus far treated T above and below Tc separately and have thereby introduced the distinction between the low temperature exponents α , γ and ν and their high temperature counterparts α, γ and ν. However, there are no singularities for H = 0 at T = Tc so the scaling functions for T < Tc and T > Tc must smoothly and analytically match together at T = Tc for H = 0. Furthermore from the exact calculations in D = 2 for the Ising model at H = 0 we find that in fact α = α, γ = γ

ν = ν.

(5.50)

These facts will be incorporated in our scaling theory by dropping the distinction between high and low temperature exponents and by extending the definition of the scaled magnetic field to H h = lim (5.51) H→0 |T − T |ν(D+2−η)/2 c T →Tc which is now taken to hold both above and below Tc . The distinction between T > Tc and T < Tc may be avoided if we define, in analogy to the scaled field h, a scaled temperature 2 τ = lim (T − Tc )H − ν(D+2−η) . (5.52) H→0 T →Tc

We now may define 2

κ = |T − Tc |ν + H D+2−η and in terms of (5.53) we define a general scaling function

(5.53)

½


G(r, τ ) = AG lim κ−(D−2+η) σ0 σR scaling

(5.54)

where AG is independent of T and H but may depend on the lattice interaction constants, and where the scaling limit is defined as T → Tc , H → 0,

R→∞

(5.55)

both fixed

(5.56)

with r = AR κR and τ

where AR depends on the lattice constants but is independent of T and H. Scaling theory assumes that the behavior of σ0 σR near T = Tc and H = 0 is given in terms of the scaling function G(r, τ ) defined by (5.54), as σ0 σR ∼ κD−2+η A−1 G G(κR, t)

(5.57)

and that the lattice-dependent constants AG and AR can be chosen such that the scaling function G(r, τ ) is independent of the lattice constants. This definition of the scaling function will reduce to the previous definitions of G± (r) if we choose the lattice-dependent constants AR and AG such that as T → Tc − AR κ ∼ 1/ξ< and M 2 (T ) ∼ AG κ2β/ν .

(5.58)

G± (r) = G(r, ±∞).

(5.59)

Thus we find We may now make contact with the thermodynamic exponent δ by noting that from (5.57) the magnetization M (T, H) is related to the scaling function G(r, τ ) by M 2 (T, H) = lim σ0 σR ∼ κD−2+η A−1 G G(∞, τ ) R→∞

(5.60)

where G(∞, τ ) is assumed to be finite. At T = Tc we have τ = 0 and thus, with the assumption that G(∞, 0) = 0, we compare the H dependence of M 2 (Tc , H) for H ∼ 0 given by (5.60) with the definition of the exponent δ of Table 5.2 to find δ=

D+2−η D−2+η

(5.61)

which is also written as

δ−1 . (5.62) δ+1 For the two-dimensional Ising model we have seen that η = 1/4, and thus (5.61) gives 2−η =D

δ = 15.

(5.63)

Finally we note that the magnetization M (T, H) and the susceptibility χ(T, H) are given in terms of the free energy as M (T, H) = −

∂ F (T, H) ∂H

(5.64)


½

and

∂ ∂2 M (T, H) = −kB T F (T, H). ∂H ∂H 2 Near T = Tc and H = 0 we write a scaling form for the free energy as kB T χ(T, H) = kB T

F (T, H) ∼ F (T, H)analytic + |T − Tc |2−α f (τ )

(5.65)

(5.66)

and use ∂ ∂τ ∂ 2 ∂ = =− |T − Tc |−ν(D+2−η)/2 |τ |1+ν(D+2−η)/2 (5.67) ∂H ∂H ∂τ ν(D + 2 − η) ∂τ where we will define for convenience what is called the gap exponent ∆ = ν(D + 2 − η)/2.

(5.68)

Thus we obtain in the scaling limit M (T, H) ∼ ∆−1 |T − Tc |2−α−∆ |τ |1+∆

∂f (τ ) ∂τ

(5.69)

and

∂ ∂f (τ ) |τ |1+∆ . (5.70) ∂τ ∂τ Now if we set H = 0 and compare the temperature dependence with the parameterizations in Table 5.2 we find the exponent relations kB T χ(T, H) = ∆−2 |T − Tc |2−2∆ |τ |1+∆

β = 2 − α − ∆ = 2 − α − ν(D + 2 − η)/2

(5.71)

γ = γ = 2 − α − 2∆ = 2 − α − ν(D + 2 − η).

(5.72)

The expression (5.71) for β will agree with the previously determined relation (5.34) 2β = D − 2 + η,

(5.73)

and the expression (5.72) will agree with (5.43) if Dν = 2 − α

(5.74)

which is the relation (5.44) with the identification of high and low temperature exponents (5.50). Unfortunately the Ising model has only been exactly solved at H = 0, and thus the only exact information we have comes from the free energy, spontaneous magnetization and susceptibility at H = 0, which is sufficient only to give the following results: f (−∞) = f (∞) = 0 ∂f (τ ) ∂f (τ ) = 0, lim τ 1+∆ = const > 0 lim τ 1+∆ τ →∞ τ →−∞ ∂τ ∂τ ∂ ∂f (τ ) = const± = 0 lim |τ |1+∆ |τ |1+∆ τ →±∞ ∂τ ∂τ where the constants const± in (5.77) are known to be quite different.

(5.75) (5.76) (5.77)

½

5.2.3


Summary of critical exponent equalities

The assumptions of scaling theory given above show that only two of the exponents are independent. These are usually taken to be the high temperature exponents for the specific heat α and for the susceptibility γ. All other exponents may be obtained by use of the scaling laws. The exponent β follows from (5.9) and (5.50) as 1 (2 − α − γ) 2 the exponent ν follows from (5.44) and (5.50) as β=

ν = (2 − α)/D

(5.78)

(5.79)

the exponent ν follows from (5.43) and (5.79) as η =2−

Dγ 2−α

(5.80)

the exponent δ follows from (5.61) and (5.80) as δ=

2−α+γ 2−α−γ

(5.81)

and the gap exponent ∆ follows from (5.68) as (5.79) and (5.80) 1 (2 − α + γ). (5.82) 2 We also recall the assumed equality of high and low temperature exponents (5.50) ∆=

α = α γ = γ ν = ν .

(5.83)

Thus, for Ising-like systems, the scaling laws compute all exponents, including the exponent of spontaneous magnetization which is only defined for T < Tc , in terms of the high temperature exponents α and γ.

5.3

Scaling for general systems

The scaling theory developed for the Ising model in the previous two sections began with critical exponents defined in the low temperature regime at H = 0 and, from the low temperature exponents and correlation functions, the definition of the low temperature scaling function for the two-point function (5.29) was constructed. This low temperature definition was then extended to temperatures above Tc and to include H = 0 as well. The final result was a theory which is valid in the scaling region T → Tc and H → 0. For more general systems, however, this construction which starts from the low temperature regime is not always either convenient or appropriate. Instead of attempting to construct a theory in the scaling region by beginning with a low temperature theory, we reverse the process and posit ab initio a scaling theory valid in the scaling region, and only afterwards will we attempt to make contact with results obtained for high and low temperatures. We will illustrate this for the classical n vector model, the Heisenberg magnet and the Lennard-Jones fluid.


5.3.1

½

The classical n vector and quantum Heisenberg models

The classical n vector is defined by the Hamiltonian z H = −J SR · SR − H SR R,R

(5.84)

R

where SR is a classical vector with n ≥ 2 components at the site R which satisfies S2 = 1

(5.85)

and R and R will be restricted to be nearest neighbors on some lattice. When n = 3 this is the classical Heisenberg magnet. The Hamiltonian for the quantum mechanical Heisenberg model is the same as (5.84) where now SR are spin operators that commute with each other on different sites and on the same site obey the commutation relations

with

y y y x z z x z x [SR , SR ] = iSR , [SR , SR ] = iSR , [SR , SR ] = iSR

(5.86)

S†R = SR and S2R = S(S + 1)

(5.87)

where S is an integer or half integer and is the magnitude of the quantum spin. We will by convention assume that for T < Tc the spontaneous magnetization is in the z direction. There will then be two types of correlations to consider: correlations of z i the spins SR which we will call longitudinal correlations and correlations of spins SR which we call transverse correlations. We note that, for T > Tc and H = 0, it follows from rotational invariance in spin space that z i S0z SR = S0i SR for all i.

(5.88)

However, for H = 0 and H = 0 with T < Tc equality of the longitudinal and transverse correlations does not hold. The longitudinal correlations are analogous to the Ising correlations whereas the transverse correlations are a new feature. The magnetization of the n vector model is defined as M (T, H) = S z = Z(T, H)−1 dSR S0z e−H/kB T (5.89) R

where Z(T, H) =

dSR e−H/kB T

(5.90)

R

and for the quantum Heisenberg model the similar formula holds with the integral replaced by the trace over all states. Thus the magnetic susceptibility is given in terms of the longitudinal correlations as χ (T, H) =

∂M (T, H) z = {S0z SR − M (T, H)2 } ∂H R

(5.91)

½


and in an analogous fashion we define a transverse susceptibility as x S0x SR . χ⊥ (T, H) =

(5.92)

R

Scaling theory for the Ising model is based on the assumption that on the lattice there exists a diverging correlation length ξ −1 = κ = |T − Tc |ν + H D+2−η 2

(5.93)

and that this is the only length scale (in addition to the lattice length) in the system. If we make the identical assumption for the n vector and quantum Heisenberg model we define the longitudinal and transverse scaling functions in the scaling limit

with

T → Tc , H → 0, R → ∞

(5.94)

AR κR = r and τ = (T − Tc )H − ν(D+2−η) both fixed

(5.95)

x . G⊥ (r, τ ) = AG⊥ lim κ−(D−2+η) S0x SR

(5.96)

z G (r, τ ) = AG lim κ−(D−2+η) S0z SR

(5.97)

G (∞, τ ) = AG lim κ−(D−2+η) M 2 (T, H).

(5.98)

2

as scaling

scaling

and

scaling

For the Ising model in D = 2 and H = 0 the scaling function G(r, ±∞) in the limit r → ∞ smoothly connects to the behavior of σ0 σR where |T − Tc | is small but R|T − Tc | is large. Thus the scaling of the two-dimensional Ising model is a means extending our considerations of the system away from the critical point into a region in the vicinity of T = Tc . Scaling theory for general systems assumes that this same connection principle holds in general and thus we formulate the following connection principles for the two correlation functions of the n vector model. Connection of scaling functions at r → ∞ 1). The behavior of G⊥ (r, τ ) when r → ∞ smoothly connects to the behavior of x S0x SR when R is first made large and then T − Tc and H are allowed to become small. 2). The behavior of G (r, τ ) when r → ∞ smoothly connects to the behavior of z S0z SR when R is first large and then T − Tc and H are allowed to become small. x z We thus need to inquire about the behavior of the correlations S0x SR and S0z SR for large R on the lattice. The following summarizes what is known or assumed for these behaviors and the implications for the scaling functions which follow from the connection principle.


½

Connection at r → ∞ for H = 0 and T > Tc for D = 3 The large R behavior in D = 3 of the correlations is z x S0z SR = S0x SR ∼

C> (T, Ω)e−R/ξ(T,Ω) R

(5.99)

where Ω specifies the direction of R. Connection requires that for large r G (r, ∞) = G⊥ (r, ∞) ∼

c> e−r as r → ∞ r

(5.100)

where 0 < c> < ∞

(5.101)

lim ξ(T, Ω)|T − Tc |ν = A−1 R

(5.102)

T →Tc

and AR is the lattice-dependent constant introduced in (5.95). From this connection we will be able to estimate the exponents γ and α from the high temperature series expansions we will develop in chapter 9. Connection at r → ∞ for H = 0 for D = 3 From low temperature spin wave computations for the quantum [14, 15] and classical [16, 17] Heisenberg magnet and from the inequality of Dunlop and Newman [18] for the classical n = 2 vector model x 2 z z S0x SR ≤ {S0z SR − M 2 (T, H)}{S0z SR + M 2 (T, H)}

(5.103)

it is inferred that as R → ∞ for H = 0 and all T = Tc that for D = 3 z x 2 − M 2 (T, H) ∼ M −2 (T, H)S0x SR . S0z SR

(5.104)

For large R and all T = Tc and H it is also inferred from low temperature spin wave computations that e−R/ξ⊥ (T,Ω) x (5.105) S0x SR ∼ C⊥ (T, H, Ω) R and thus from (5.104) for large R z S0z SR − M 2 (T, H) ∼ C (T, H, Ω)

e−2R/ξ⊥ (T,Ω) R2

(5.106)

where 2 (T, H, Ω). C (T, H, Ω) = M −2 (T, H)C⊥

(5.107)

Connection of these two expressions with the definitions of the scaling limit (5.96) and (5.97) requires that C⊥ (T, H, Ω) ∼ κη

(5.108)

½


C (T, H, Ω) ∼ κ−1+η

(5.109)

and thus using (5.105), (5.106), (5.108) and (5.109) and the κ dependence of the scaling law for M (T, H) (5.69) with (5.74) M 2 (T, H) ∼ κ1+η

(5.110)

κ2η ≤ κ2η

(5.111)

in (5.103) we find which is obviously satisfied. We thus see for τ finite and r large that the scaling functions behave as G⊥ (r, τ ) ∼ c⊥ (τ )

e−κ⊥ (τ )r r

G (r, τ ) ∼ G (∞, τ ) + c (τ )

(5.112) e−2κ⊥ (τ )r r2

(5.113)

with 0 < c⊥ (τ ), G (∞, τ ), c (τ ) < ∞

(5.114)

where κ⊥ (τ ) is a new correlation length which satisfies 0 ≤ κ⊥ (τ ) < ∞

(5.115)

for all τ including ±∞. Connection at r → ∞ for H = 0 and T < Tc for D = 3 When H → 0 it is known from low temperature spin wave computations [19, 20] that ξ⊥ (T, H, Ω) → ∞ and thus the large R behavior of the correlations is C⊥ (T, 0, Ω) R C (T, 0, Ω)

z S0z SR − M 2 (T, 0) ∼ . 2 R x S0x SR ∼

(5.116) (5.117)

Connection requires that for large r G⊥ (r, −∞) ∼

c⊥ (−∞) r

G (r, −∞) ∼ G (∞, −∞) +

(5.118) c (−∞) r2

(5.119)

and thus we must have κ⊥ (−∞) = 0

(5.120)

Connection of scaling functions at T = Tc and H = 0 The scaling function G(r, τ ) of the Ising model is also assumed to have the property that, when r → 0, there is a smooth connection to the large R behavior of σ0 σR


½

when T = Tc . For the n vector model this connection requires that at T = Tc and H = 0 for R → ∞ const z x S0z SR = S0x SR ∼ D−2+η (5.121) R and that for r → 0 G G⊥ (r, τ ) = G (r, τ ) ∼ D−2+η (5.122) r where G is a finite nonzero constant which is independent of τ. For the Ising model in D = 2 this is confirmed for τ = ±∞ but no independent confirmation at finite τ yet exists. Diverging susceptibility at H = 0 for T < Tc The transverse and longitudinal susceptibilities are expressed in terms of the correlations by (5.91) and (5.92) which when we use forms (5.96) and (5.97) written as x D−2+η S0x SR ∼ A−1 G⊥ (κr, τ ) G⊥ κ z S0z SR

− M (T, H) ∼ 2

D−2+η A−1 {G (κr, t) G κ

(5.123) − G (∞, τ )}

(5.124)

become in the scaling limit χ⊥ (T, H) ∼

−2+η A−1 G⊥ κ

−2+η χ (T, H) ∼ A−1 G κ

dD rG⊥ (r, τ )

(5.125)

dD r{G (r, t) − G(∞, τ )}.

(5.126)

If for H = 0 and T > Tc we use the asymptotic forms (5.100) and (5.122) we see that the integrals converge and thus find that the susceptibility exponent satisfies the exponent relation γ = ν(2 − η). However, for H = 0 and T < Tc we see from the asymptotic forms that the integrals in both (5.125) and (5.126) diverge. Thus the susceptibilities χ⊥ (T, H) and χ (T, H) for T < Tc both diverge as H → 0 and thus the low temperature exponent γ does not exist. The divergence of χ (T, H) was first seen by Dyson [19,20] in 1956 where he showed for the quantum Heisenberg magnet in three dimensions that for T ∼ 0 χ (T, H) ∼ const H −1/2 .

(5.127)

The computation is done in terms of “spin wave excitations” and is one of the classic computations of magnetism in condensed matter physics. For the classical n = 2 vector model it was shown rigorously by Lebowitz and Penrose [21] in 1975 from the inequality (5.103) but without using scaling that χ (T, H) ≥ const M (T, H)7/2 H −1/2 for D = 3 const for D = 4 const M (T, H)4 ln HM (T, H) which diverge as H → 0 for T < Tc where by definition M (T, 0) > 0.

(5.128)

½


These divergences for H → 0 are obtained by using the forms (5.112) and (5.113) in the integrals of (5.125) and (5.126). Thus we find ∞ −1 −2+η −2 χ⊥ (T, H) ∼ AG⊥ c⊥ (−∞)κ κ⊥ 4π dy ye−y (5.129) 0 ∞ −2+η −1 χ (T, H) ∼ A−1 κ⊥ 4π dy e−y . (5.130) G c (−∞)κ 0

This will give a divergence of H −1/2 in D = 3 for χ if as t → −∞ −1/2 . κ−1 ⊥ (τ ) ∼ H

(5.131)

Thus, since κ⊥ (τ ) is a function of τ alone we find from (5.95) that as τ → −∞ with D=3 ν(5−η)/4 κ−1 = H −1/2 (Tc − T )ν(5−η)/4 (5.132) ⊥ (τ ) ∼ τ and hence we find the final results [17] ν(1−η)/2 −1 H χ⊥ (T, H) ∼ A−1 G⊥ 4πc⊥ (−∞)(Tc − T )

χ (T, H) ∼ AG 5.3.2

−1

−ν3(1−η)/4

4πc (−∞)(Tc − T )

H

(5.133) −1/2

.

(5.134)

Lennard-Jones fluids

The Lennard-Jones fluid introduced in chapter 2 is a classical continuum model with a two body interaction potential with n > m U (R) = (R/σ)−n − (R/σ)−m .

(5.135)

This potential decays as (R/σ)−m as R → ∞, and consequently the two-particle correlation can decay as R → ∞ no faster than R−m . This algebraic decay means that we cannot define a correlation length ξ(T ) from the decay e−R/ξ(T ) and hence we cannot define the exponent ν from the divergence of ξ(T ) as T → Tc . Nevertheless we can still formally define a κ from the fluid analogue of (5.53) of the Ising model definition and scaling functions from the fluid analogue of (5.54). However, now the large r behavior of the scaling function will have to match for the power law behavior determined by the R−m . The scaling laws for exponents will still formally hold but there are no high temperature expansions from which the exponents can be estimated.

5.4

Universality

We conclude this chapter with the introduction of the concept of universality. Loosely speaking this is the idea that near the critical point when the correlation length becomes large compared with the lengths over which the potential is nonzero that there should be properties of the behavior which do not depend on the details of the interaction but only on properties such as the dimensionality of space and the number of components in the n vector model. The first place such a universality can be seen is in the comparison of the exact solution of the Ising model on the triangular and square lattice where all the dependence

Missing theorems

½

on the lattice constants is in the factors AR and AG of (5.95), (5.96), and (5.97). More generally, all the evidence is that, for any finite range interactions in an Ising system in either two or three dimensions, it is still the case that all dependence on the lattice interactions is contained in the AR and AG . A less obvious statement of universality relates to the relation between the quantum and classical Heisenberg model. The weakest version of universality for these systems is the statement that the critical exponents do not depend on the quantum spin S of the system and thus that the classical and quantum Heisenberg magnets have the same exponents. Moreover for the classical Heisenberg magnet the ferromagnet and the antiferromagnet map onto each other if the lattice is bipartite, merely by changing the sign of the spin on one sublattice. Therefore universality will say that the critical exponents of the quantum ferromagnet and antiferromagnet are the same. This is a striking assertion because we saw in the previous chapter that while spontaneous antiferromagnetic order could be proven to exist in the three-dimensional quantum Heisenberg antiferromagnet there is no proof of order for the quantum ferromagnet. Clearly the concept of universality and scaling, if correct, is in advance of our ability to make rigorous proofs. It can also be asked if, for the Heisenberg magnet, the complete scaling function is independent of the quantum spin. The answer to this seems not to be known. Finally we would like to know if there is any universality between the threedimensional lattice gas (which is isomorphic to the three-dimensional Ising model) and the Lennard-Jones fluid which is thought to describe real inert gases. It is obvious that the scaling functions are not the same, but it is commonly asserted that universality implies that the critical exponents of the three-dimensional Ising model are the same as the Lennard-Jones fluid and as real inert gases.

5.5

Missing theorems

When scaling theory is defined by (5.94)–(5.98) the predictive power of scaling comes from the assumption that the connection constants c> , c⊥ (τ ), c (∞, τ ) and c (τ ) of (5.100), (5.112), and (5.113) are finite and nonzero for all τ including ±∞, and G of (5.122) is finite, nonzero and independent of τ . The only case for which this has been proven true is the Ising model in two dimensions at H = 0. These scaling assumptions allow us to predict the low temperature exponent β of the spontaneous magnetization and the exponent η of the correlations functions at T = Tc from the high temperature exponents α and γ, and these two exponents can be estimated by means of high temperature series expansions. From universality we predict that these exponents depend only on the dimension D of space and the number of components n of the spin variables (or order parameter) and are independent of all other details of the interaction energies if they are not too long range. Scaling has powerful support from the fact that all the predictions are proven exactly true for the two-dimensional Ising model at H = 0. Universality is supported by the fact that the two-dimensional Ising model can be exactly solved on the triangular lattice with three different values of the interaction strengths on the three legs of

½


the triangle and when the interactions are all positive the exponents and the scaling functions are independent of the interaction constants. These observations were the starting point for the development of scaling and universality. However, these results were obtained by using the very strong integrability properties of the Ising model, and thus it may be asked whether for models which do not have these strong integrability properties all of the predictions of scaling and universality will continue to hold. Because of the striking and powerful predictions of scaling and universality it would be most desirable to have some proofs of their validity which do not rely on exact integrability of the system. Unfortunately no such theorems exist at present. We list in Table 5.5 a few of these “missing theorems” which have arisen in the course of this chapter. Table 5.5 Some “missing theorems” in the theory of scaling and universality.

1. Prove the assumed T = Tc connection (5.122) for A. The Ising model in D = 2 for H = 0 B. The Ising model in D = 3 C. The n = 3 classical Heisenberg model D. The n = 3 spin S quantum Heisenberg ferromagnet. 2. Prove that the exponents of the spin S quantum Heisenberg ferromagnet are independent of S. 3. Prove that the exponents of the quantum Heisenberg antiferromagnet are the same as those of the ferromagnet 4. Prove that the exponents of the Ising and/or Heisenberg ferromagnet. are equal for the cubic, fcc and bcc lattices 5. Prove the inequality (5.103) for the n = 3 component classical Heisenberg ferromagnet. 6. Prove the inequality (5.103) for the spin S quantum Heisenberg ferromagnet.

References [1] M.E. Fisher, The susceptibility of the plane Ising model, Physica 25 (1959) 521– 524. [2] G.S. Rushbrooke, On the thermodynamics of the critical region for the Ising problem, J. Chem. Phys. 39 (1963) 842–843. [3] J.W. Essam and M.E. Fisher, Padé approximant studies of the lattice gas and Ising ferromagnet below the critical point, J. Chem. Phys. 38 (1963) 802–812. [4] B. Widom, Equation of state in the neighborhood of the critical point, J. Chem. Phys.43 (1964) 3808–3905. [5] M.E. Fisher, Correlation functions and the critical region of simple fluids, J. Math. Phys. 5 (1964) 944–962. [6] R.B. Griffiths, Thermodynamic inequality near the critical point for ferromagnets and fluids, Phys. Rev. Letts. 14 (1965) 623–624. [7] R.B. Griffiths, Ferromagnets and simple fluids near the critical point: some thermodynamic inequalities, J. Chem. Phys. 43 (1965) 1958–1968. [8] G.S. Rushbrooke, On the Griffiths inequality at a critical point, J. Chem. Phys. 43 (1965) 3439–3441. [9] L.P. Kadanoff, Scaling laws for Ising models near Tc , Physics 2 (1966) 263–272. [10] M. E. Fisher, The theory of equilibrium critical phenomena, Reports in Progress in Physics, 36 (1967) 615–730. [11] L.P. Kadanoff, W. G¨ otze, D. Hamblen, R.Hecht, E.A.S. Lewis, V.V. Paliauskas, M. Rayl, J. Swift, D. Aspenes and J. Kane, Static phenomena near critical points: Theory and experiment, Rev. Mod. Phys. 39 (1967) 395–431. [12] M.J. Buckingham and J.D. Gunton, Correlations at the critical point of the Ising model, Phys. Rev. 178 (1969) 848–853. [13] M.E. Fisher, Rigorous inequalities for critical-point correlation exponents, Phys. Rev. 180 (1969) 594–600. [14] R.B. Stinchcombe, G. Horwitz, F. Englert and R. Brout, Thermodynamic behavior of the Heisenberg ferromagnet, Phys. Rev. 130 (1963) 155–176. [15] R. Silberglitt and A.B. Harris, Dynamics of the Heisenberg magnet at low temperatures, Phys. Rev 174 (1968) 640–658. [16] A. Patashinskii and Z. Pokrovsky, Longitudinal susceptibility and correlations in degenerate systems, Sov. Phys. JETP 37 (1973) 733–736. [17] M.E. Fisher, M.N. Barber and D. Jasnow, Helicity modulus, superfluidity and scaling in isotropic systems, Phys. Rev. A8 (1973) 1111–1124. [18] F. Dunlop and C.M. Newman, Multicomponent field theories and classical rotators, Comm. Math. Phys. 44 (1975) 223–235. [19] F.J. Dyson, General theory of spin wave interactions, Phys. Rev. 102 (1956) 1217– 1230.

½

References

[20] F.J. Dyson, Thermodynamic behavior of an ideal ferromagnet, Phys. Rev. 102 (1956) 1230–1244. [21] J.L. Lebowitz and O. Penrose, Divergent susceptibility of isotropic ferromagnets, Phys. Rev. Letts. 35 (1975) 549–552.

Part II Series and Numerical Methods ...if one scheme of happiness fails, human nature turns to another, if the first calculation is wrong, we make a second better. Jane Austen


6 Mayer virial expansions and Groeneveld’s theorems In this chapter we study the expansion of the equation of state when the density ρ = 1/v = N/V is small for a system of N particles in a volume V of mass m interacting through the pair potential U (r) with the Hamiltonian H=

N p2i + 2m i=1

U (ri − rj ).

(6.1)

1≤i<j≤N

These expansions were first derived by Mayer and Mayer [1]. The most elementary result is that as v → ∞ Pv 1 dD r(e−U (r)/kB T − 1) + O(v −2 ). =1− kB T 2v

(6.2)

The coefficient of 1/v is called the second virial coefficient. This will be derived in section 6.1. In order to systematically extend (6.2) we introduce what is called the Mayer function f (ri,j ) ≡ fi,j = e−U (ri −rj )/kB T − 1, (6.3) to write exp{−

1 kB T

U (ri − rj )} =

1≤i<j≤N

(1 + fi,j )

(6.4)

1≤i<j≤N

and expand the product. The expansion (6.4) consists of 2N terms, each of which can be represented in what is called a Mayer graph, by a set of points numbered from 1 to N and m lines representing the functions fi,j where m ranges from zero to 2N (N −1)/2 An example of a Mayer graph with ten points is given in Fig. 6.1. In general these graphs may consist of a number of disconnected pieces. In the grand canonical ensemble in a finite volume V we will denote the pressure and density as a function of the activity (or fugacity) z by p(z; V ) and ρ(z; V ) respectively and refer to their expansions for small z as the activity (or cluster) expansion. It is given by Mayers’ first theorem, which will be proven in section 6.2, as ∞

p(z; V ) k = z bk (V ) kB T k=1

(6.5)

½

Mayer virial expansions and Groeneveld’s theorems

Fig. 6.1 An example of a Mayer graph with 10 points. This graph has four disconnected pieces and its value is f1,6 f2,3 f3,8 f4,9 f5,9 f9,10 f5,10 f4,5 f5,10 ∞

ρ(z; V ) = z

d (p(z; V )/kB T ) = kz k bk (V ) dz

(6.6)

k=1

where

1 bk (V ) = k!V

Uk (r1 , · · · , rk )dD r1 · · · drD k

(6.7)

V

and Uk (r1 , · · · , rk ) is the sum of all connected numbered (labeled) Mayer graphs of k points. The quantities bk (V ) are known as the cluster integrals and the functions Uk (r1 , · · · , rk ) are known as the Ursell functions. In section 6.3 an expression in infinite volume is derived for the pressure P (ρ) in terms of the (average) density ρ = limV,N →∞ N/V P (ρ) =ρ+ Bk+1 ρk+1 kB T

(6.8)

k=1

with

k βk k+1

(6.9)

Vk+1 (r1 , · · · , rk+1 )dD r2 · · · dD rk+1 ,

(6.10)

Bk+1 = − where βk =

1 k!

the integrals are over all (infinite) space and Vk (r1 , · · · , rk ) are those connected numbered Mayer diagrams with k points which do not become disconnected by the removal of any one point. Such Mayer graphs are called either biconnected or irreducible and the functions Vk (r1 , · · · , rk ) are called Husimi functions. The expansion (6.8) is called the virial expansion, the βk are known as the irreducible cluster integrals, the Bk are called the virial coefficients and the derivation is known as Mayer’s second theorem. In the appendix we give the unlabeled irreducible (biconnected) graphs of four and five points and the number of labeled graphs to which they correspond. The six-point graphs are given in [2]. It is also possible to invert (6.6) to obtain z(ρ; V ) and use this in (6.5) to obtain a third expansion P (ρ; V ) p(z(ρ; V )) = =ρ+ Bk+1 (V )ρk+1 kB T kB T k=1

(6.11)


with Bk+1 (V ) =

k βk (V ) k+1

½

(6.12)

but unlike the bk (V ) defined by (6.7), even though the βk (V ) are given as polynomials in the bk (V ), these βk (V ) are not given by restricting the integrals in (6.10) to a finite volume. This expansion will be considered in section 6.5. These expansions are useful whenever they converge, and we will consider four separate cases. We call R(V ) the radius of convergence of (6.5) and (6.6) and thus the radius of convergence of the limiting case V → ∞ of the series ∞

p(z; V ) = lim z k bk (V ), V →∞ kB T V →∞ lim

lim ρ(z; V ) = lim

V →∞

k=1

V →∞

∞

kz k bk (V )

(6.13)

k=1

is lim R(V ).

V →∞

(6.14)

It follows from the discussion of first order phase transitions and grand partition function zeros in 3.4 that there are no phase transitions for |z| ≤ lim R(V ). V →∞

(6.15)

We call R the radius of convergence of the series defined as ∞

p(z) , kB T

z k bk =

k=1

with bk = lim bk (V ) = V →∞

1 k!

∞

kz k bk = ρ(z)

(6.16)

Uk (r1 , · · · , rk )dD r2 · · · dD rk

(6.17)

k=1

where the integrals are over space. The expansion (6.17) differs from (6.13) in that all ∞ the limV →∞ and the sum k=1 have been interchanged. In general lim R(V ) ≤ R

V →∞

(6.18)

which expresses the fact that the limiting position of the zero of Qgr (z, T ; V ) closest to the origin does not have to lead to a singularity of the pressure in the thermodynamic limit. We will similarly define R to be the radius of convergence of (6.8) and R(V ) to be the radius of convergence of (6.11). An argument similar to the argument given above shows that in general lim R(V ) ≤ R (6.19) V →∞

and that the system will not have any phase transitions for |ρ| ≤ lim R(V ). V →∞

(6.20)

½


In section 6.4 we prove the theorems of Groeneveld [3] on the radius of convergence R for non-negative potentials U (r) ≥ 0. (6.21) In particular we will first prove that 0 ≤ (−1)k−1 bk (V ) ≤ (−1)k−1 bk

(6.22)

bk k k−2 1 ≤ . ≤ k−1 k (2b2 ) k!

(6.23)

and

When k = 2 the upper and lower bounds are both equal to 1/2. From (6.23) it will follow that R 1 1 −N Φ (6.42) 1≤i<j≤N

and for which the integral

dD r|e−U (r)/kB T − 1|

C=

(6.43)

V

converges. For these potentials the radius of convergence of the cluster expansion in a finite volume satisfies 1 l bl (V )|1/(l−1) ≤ R(V ) ≤ |eΦ/kB T l−1 Ce1+2Φ/kB T

(6.44)

for any l. The lower bound in (6.44) was first shown by Ruelle [8] and the upper bound was first shown by Penrose [9]. Lebowitz and Penrose [4] have shown that the the radius of convergence R of the virial expansion satisfies R ≥ lim R(V ) ≥ V →∞

0.14476 2 C 1+u

(6.45)

where u = e2Φ/kB T .

(6.46)

For non negative potentials u = 1 and C = 2B2 , and thus (6.45) reduces to (6.29). The best results for βk are those of Groeneveld [7, part IV eqns.(3.30),(3.31)] |βk | ≤ where

Ck pk−1 (1 + u) k

1 pj (y) = 2πi

dξ (yeξ − 1)j ξ j+1

(6.47)

(6.48)

and the more refined bounds stated (but not proven) in [7, part IV eqns.(3.34),(3.35)] βk− ≤ βk ≤ βk+ where for k ≥ 2

(6.49)


βkµ = µ

Ck (2B2 )k {pk−1 (1 + u) − 1} − {pk−1 (1 − u) + 1}. 2k 2k

½

(6.50)

By use of the inequality which can be derived from (6.48) kk k!

(6.51)

(k − 1)k−1 . k!

(6.52)

pk (1 + u) ≤ (1 + u)k we obtain from (6.47) [7, part IV eqn.(3.39)] |βk | ≤ C k (1 + u)k−1

An inequality stronger than (6.51) for large k is [7, part IV eqn.(3.41)] pk (1 + u) ≤

1 c(u)k

(6.53)

where c(u) is the smallest root of the equation [7, part IV eqn.(3.42)] c(u)e−c(u) =

1 . (1 + u)e

(6.54)

Thus using (6.53) in (6.47) we find [7, part IV eqn.(3.43)] |βk | ≤

1 Ck k c(u)k−1

(6.55)

and thus the radius of convergence of the virial expansion R satisfies R≥

c(u) . C

(6.56)

Finally, using c(u) =

ec(u) 1 ≥ (1 + u)e (1 + u)e

(6.57)

in (6.56) we obtain [7, part IV eqn.(5.46)] R≥

0.18394 · · · 2 2 1 = 2eC 1 + u C 1+u

(6.58)

which is stronger that (6.45). The convergence of the integral C (6.43) is guaranteed for weakly tempered potentials, and thus the lower bounds (6.45) and (6.58) prove that the virial expansion has a finite radius of convergence for all stable weakly tempered potentials. We finally conclude in section 6.6 with theorems on the counting of Mayer diagrams.

½


6.1

The second virial coefficient

In the canonical ensemble the pressure is given as ∂A P =− ∂V T

(6.59)

where A, the Helmholtz free energy, is defined from the partition function 1 dD p1 · · · dD pN dD r1 · · · dD rN e−H/kB T QN (V, T ) = N! V

(6.60)

as A = −kB T lnQN (V, T ).

(6.61)

For the class of Hamiltonians given by (6.1) the Gaussian integrals over the variable pi are easily done, and thus (6.60) is more explicitly written as 1 − k 1T U (ri −rj ) D D i<j B QN (V, T ) = d r · · · d r e (6.62) 1 N N !λDN V where λ, which is called the thermal wavelength, is defined by λ=

1 . (2πmkB T )1/2

(6.63)

We are interested in the thermodynamic limit where V → ∞, N → ∞, with ρ =

N 1 = fixed. v V

(6.64)

In the noninteracting case where U (r) = 0 we trivially have QN (V, T ) =

VN . N !λDN

(6.65)

Therefore from (6.61) we have A = −kB T (N lnV − ln(N !λDN ))

(6.66)

and thus from (6.59) kB T N kB T = V v which is the equation of state for the ideal gas. In general we write VN ˜ QN (V, T ) QN (V, T ) = N !λDN where − k 1T U (ri −rj ) D D ˜ N (V, T ) = 1 i<j B Q d r · · · d r e 1 N VN V P =

(6.67)

(6.68)

(6.69)

The second virial coefficient

½

and define

1 ˜ (6.70) lnQN (V, T ) N under the assumption that, in the thermodynamic limit (6.64), the limit on the righthand side exists. Then, using (6.70) in (6.68) and (6.61) we find from (6.59) f˜(v, T ) =

lim

N →∞, V →∞

1 ∂ P = + f˜(v, T ) kB T v ∂(V /N )

(6.71)

and thus

∂ Pv = 1 + v f˜(v, T ). kB T ∂v To proceed further we use (6.4) in (6.69) to write 1 ˜ QN (V, T ) = N dD r1 · · · dD rN V V

(6.72)

(1 + fi,j )

(6.73)

1≤i<j≤N

and expand the product. In order for the limit (6.70) to exist it is necessary that ˜ N (V, T ) behave as an exponential in N as N → ∞ and this is achieved in the lowest Q order of approximation by keeping only those terms in the expansion of the form fj1 ,j2 fj3 ,j4 · · · fj2n−1 ,j2n

(6.74)

where all of the ji are distinct. For a given n the number of such terms is N (N − 1)(N − 2) · · · (N − 2n + 1) n!2n

(6.75)

and thus we have n N (N − 1) · · · (N − 2n + 1) 1 D d rf (r) . 1,2 n!2n V V n=1 N/2

˜ N (V, T ) ∼ 1 + Q

(6.76)

In the limit (6.64) this becomes n Nn 1 1 D D ˜ d rf1,2 (r) d rf1,2 (r) . (6.77) = exp N QN (V, T ) ∼ 1 + n! 2v 2v n=1 Therefore from (6.70) 1 f˜(v, T ) ∼ 2v

dD rf1,2 (r)

(6.78)

and from (6.72) we find

1 Pv dD rf1,2 (r) ∼1− (6.79) kB T 2v which is the desired result (6.2). Under the weakly tempered assumption of the pair potential (3.22) this integral converges at r → ∞. Comparing with the definition of the virial series (6.8) we see that the second virial coefficient is 1 B2 (T ) = − dD rf1,2 (r) (6.80) 2 and we note that B2 (T ) may be positive, negative or even zero. The temperature (if any) TB at which B2 (TB ) = 0 is called the Boyle temperature.

½

6.2


Mayers’ first theorem

In the derivation of the previous section it is not particularly clear that we are obtaining an expansion in terms of 1/v. Equally unclear is how to systematically obtain the higher terms in this expansion. In this section we will begin to answer these questions by using the grand canonical ensemble to prove what is called Mayers’ first theorem given by (6.5)–(6.7). The grand partition function Qgr (z, T ; V ) is defined as Qgr (z, T ; V ) =

∞

(λD z)N QN (V, T )

(6.81)

N =0

and from this we have

1 p(z; V ) = lnQgr (z, T ; V ) kB T V

(6.82)

and

1 ∂ z lnQgr (z, T ; V ). (6.83) V ∂z Further we define WN (r1 , · · · , rN ) to be the collection of all Mayer graphs of N numbered points and write (6.62) as 1 QN (V, T ) = dD r1 · · · dD rN WN (r1 · · · rN ). (6.84) N !λDN V ρ(z; V ) =

Our first step in proving (6.5)–(6.7) is to express WN +1 in terms of UN +1 , and Wk and Uk with k ≤ N . To do this we single out the point N + 1 for special attention and write WN +1 (r1 , · · · , rN +1 ) in terms of how many points the point rN +1 is connected to. For brevity we represent the coordinate rk by the subscript k. Then we have the following recursion relation [10, (7.12)] WN +1 (1 · · · N + 1) = UN +1 (1 · · · N + 1) + UN (1 · · · ˆj · · · N + 1)W1 (j) j≤N

+

UN −1 (1 · · · ˆj · · · kˆ · · · N + 1)W2 (j, k)

1≤j 0 U2 (1, 2) = f1,2 ≤ 0 T1 (1; 2) = f1,2 ≤ 0.

(6.140)

We thus may make the induction hypothesis that for integer L for l ≤ L − 1 (−1)l−1 Ul (1, · · · , l) ≥ 0 (−1)l Tl (1; 1, · · · , l) ≥ 0 L−1

(6.141)

Therefore the sign of TL−k Uk is (−1) and we find from (6.97) that the sign of UL is (−1)L−1 and thus the sign of TL is (−1)L and hence the hypothesis (6.141)

½


holds for l = L. From the definition (6.7) the sign of bk (V ) is the same as the sign of Uk (1, · · · , k). Moreover the integrand in (6.7) never changes sign. Therefore the desired result (6.7) follows that 0 ≤ (−1)k−1 bk (V ) ≤ (−1)k−1 bk .

(6.142)

B. The upper and lower bounds on bl We begin the proof of (6.23) by introducing the notation [3] f (1; 2, · · · , l) =

l

(1 + f1,j ) − 1

(6.143)

j=2

and write (6.137) as Tl (1; 2, · · · , l + 1) = f (1; 2, · · · , l + 1)Ul (2, · · · , l + 1).

(6.144)

We note that l

(1 + f1,k ) = (1 + f1,l )

k=2

l−1

(1 + f1,k ) = f1,l

k=2

l−1

(1 + f1,k ) +

k=2

l−1

(1 + f1,k ).

(6.145)

k=2

Therefore by induction l

(1 + f1,k ) =

k=2

l k=3

f1,k

k−1

(1 + f1,j ) + (1 + f1,2 )

(6.146)

j=2

and thus we have f (1; 2, · · · , l) = f1,2 +

l

f1,k

k−1

(1 + f1,j ).

(6.147)

j=2

k=3

We now take the absolute value of this expression to obtain an upper bound |f (1; 2, · · · , l)| ≤ |f1,2 | +

l

|f1,k |

k=3

k−1

|1 + fi,j |

(6.148)

j=2

from which, using the fact, that for nonnegative potentials (6.136), fi,j ≤ 0 and hence that |1 + fi,j | ≤ 1, we find |f (1; 2, · · · , l)| ≤

l

|f1,k |.

(6.149)

k=2

Furthermore a lower bound on |f (1; 2, · · · , l)| is obtained if we retain only the first term in the sum (6.147). Thus we have |f1,2 | ≤ |f (1; 2, · · · , l)| ≤

l

|f1,k |.

(6.150)

k=2

We now integrate the variables r2 , · · · , rl+1 in (6.144) over all space and take the absolute value to write

Non-negative potentials and Groeneveld’s theorems

½

f (1; 2, · · · , l + 1)Ul (2, · · · , l + 1)dD r2 · · · dD rl+1

|l!tl | = =

|f (1; 2, · · · , l + 1)||Ul (2, · · · , l + 1)|dD r2 · · · dD rl+1

(6.151)

where the equality in the last line holds because of the condition of the signs of Tl and Ul previously established and now we may use the bounds of |f (1; 2, · · · , l)| (6.150) to obtain |f1,2 ||Ul (2, · · · , l + 1)|dD r2 · · · dD rl+1 ≤ |l!tl | ≤

l+1

|f1,k ||Ul (2, · · · , l + 1)|dD r2 · · · dD rl+1 .

(6.152)

k=2

Now because the point 1 is not in common with any of the points in Ul (2, · · · , l + 1) the integral over the product f1k and Ul (2, · · · , l + 1) factorizes into the product of the integrals and we obtain D |f1,2 |d r2 |Ul (2, · · · , l + 1)|dD r3 · · · dD rl+1 ≤ |l!tl | ≤ l|f1,2 |dD r2 |Ul (2, · · · , l + 1)|dD r3 · · · dD rl+1 . (6.153) and thus, using the definition of bl (6.17), we have upper and lower bounds for |tl | 2|b2 ||bl | ≤ |tl | ≤ 2|b2 |l|bl |.

(6.154)

In order to use the bounds (6.154) to find bounds on bl we note that, because of the alternation of signs of tl and bl , we may rewrite (6.101) in terms of absolute values as l−1 l(l − 1)|bl | = k|bk |(l − k)|tl−k | (6.155) k=1

in which it is convenient to set for l ≥ 1 l|bl | = al−1

(6.156)

to get lal =

l−1

am (l − m)|tl−m |.

(6.157)

m=0 L We now define the quantities aU l and al as the solutions of the recursion relations identical in form to (6.157)

laU,L = l

l−1 m=0

U,L aU,L m (l − m)tl−m

(6.158)

½


with the initial condition U aL 1 = a1 = a1 = B

where

tU,L l

(6.159)

are the upper and lower bounds on the |tl | given by (6.154) as L L tL l = B|bl | = Bal−1 /l

tU l

=

Bl|bU l |

=

BaU l−1

(6.160) (6.161)

and we have set B = 2|b2 |.

(6.162)

aU,L l

defined by the recursion relations (6.158) actually We wish to prove that the have the properties of being upper and lower bounds U aL l ≤ al ≤ al .

(6.163)

This bounding property is satisfied for l = 1 by the initial conditions (6.159). For l > 1 we prove (6.163) by induction by noting that if (6.163) is assumed to hold up to l − 1 that l−1 l−1 U U lal = am (l − m)|tl−m | ≤ aU (6.164) m (l − m)tl−m = lal m=0

and lal =

l−1

m=0

am (l − m)|tl−m | ≥

m=0

l−1

L L aL m (l − m)tl−m = lal

(6.165)

m=0

and thus (6.163) holds at l. We now define the generating functions AU,L (z) = T U,L (z) =

∞ l=0 ∞

aU,L zl, l

(6.166)

tU,L zl. l

(6.167)

l=0

Then the identical procedure which leads to (6.108) gives AU,L (z) = exp(T U,L (z)).

(6.168)

On the other hand we may use (6.160) and (6.161) in the definition (6.167) to find a second relation between T U,L (z) and AU,L (z). For the upper bound we directly substitute (6.161) into (6.167) to find T U (z) = BzAU (z).

(6.169)

For the lower bound we substitute (6.160) into (6.167) to first obtain T L (z) =

∞ l=1

zl

B L a l l−1

(6.170)

Non-negative potentials and Groeneveld’s theorems

½

from which we obtain the differential equation ∂T L(z) = BAL (z). ∂z

(6.171)

It remains to solve (6.169) and (6.171) with (6.168) to compute aU,L . l The lower bound We consider first the lower bound. Then from (6.168) we find ∂AL (z) ∂T L (z) = AL (z) ∂z ∂z

(6.172)

which, if we eliminate ∂T L(z)/∂z using (6.171), gives the differential equation ∂AL (z) = B[AL (z)]2 . ∂z

(6.173)

AL (0) = 1

(6.174)

Noting the initial condition which follows from T L (0) = 0 we solve (6.173) to obtain AL (z) =

1 . 1 − Bz

(6.175)

Therefore recalling the definition of AL (z) (6.166) we find l aL l =B

(6.176)

and recalling the definition (6.156) |bL l |=

B l−1 . l

(6.177)

Thus we have the lower bound B l−1 ≤ (−1)l−1 bl l

(6.178)

or recalling (6.162) and the sign alternation property (6.142) we obtain the final result bl 1 ≤ . l (2b2 )l−1

(6.179)

½


The upper bound For the upper bound we substitute (6.169) into (6.168) to obtain AU (z) = eBzA

U

(z)

(6.180)

or, taking the logarithm, z=

lnAU (z) . BAU (z)

(6.181)

The desired coefficients aU l can be obtained from (6.181) by using the contour integral expression 1 AU (z) aU = dz l+1 (6.182) l 2πi C z where C is a closed contour enclosing z = 0 to recover the coefficient aU l from the generating function (6.166). From (6.181) we find dz = (1 − lnAU )

dAU . BAU 2

(6.183)

Then setting AU = eξ we have ξ Beξ 1−ξ dz = dξ Beξ z=

(6.184)

and note that the closed contour C in the z plane will map into a closed contour C enclosing ξ = 0 in the ξ plane. Thus we find 1 l 1−ξ B = dξ l+1 eξ(l+1) (6.185) aU l 2πi ξ C which is readily computed by evaluating the residues at ξ = 0 to give aU l =

B l (l + 1)l−1 . l!

(6.186)

Therefore, recalling the definitions of aU l (6.156) and B (6.162) and the sign alternation property (6.142), we obtain the desired upper bound bl ll−2 . ≤ l−1 (2b2 ) l!

(6.187)

C. The bounds on the radius of convergence The radius of convergence R of the series (6.16) is obtained by applying the root test to the bounds of bl . The upper bound RU is found from the lower bound bL l as −1/l |2b2 |l−1 1 −1/l (6.188) = lim = RU = lim (|bL l |) l→∞ l→∞ l |2b2 | where liml→∞ l1/l = 1 has been used.

Convergence of virial expansions

The lower bound RL is found from the upper bound bU l as 1/l l! 1 −1/l lim |) = RL = lim (|bU l l→∞ 2|b2 | l→∞ ll−2 1 = 2e|b2 |

½

(6.189)

where, in the last line, Stirling’s approximation l! ∼ ll+1/2 e−l (2π)1/2 for l → ∞

(6.190)

has been used. Thus the bounds (6.24) on the radius of convergence have been proven.

6.5


In the introduction we introduced the expansion of the pressure P (ρ; V ) as a function of the density ρ in a finite volume V which is obtained by explicitly algebraically eliminating z between the equations (6.5) and (6.6) of the grand canonical ensemble in a finite volume. It is easy to see that this leads to algebraic relations between the finite volume bk (V ) which have been shown to be given by the integrals of the Ursell functions UN in a finite volume and the quantities βk (V ) defined by (6.12) which are obtained from this algebraic elimination. Examples of these relations are β1 (V ) = 2b2 (V )

(6.191)

β2 (V ) = 3b3 (V ) − 6b2 (V ) . 2

(6.192)

These algebraic relations are exactly the same as the relations which are obtained in infinite volume by using the formulas for βk as integrals over the the Husimi functions Vk but, unlike the integral expressions for bk (V ), the integral formulas for the βk do not extend to the finite volume function βk (V ) defined by these algebraic eliminations. In this section we will follow [4] to find a lower bound on the radius of convergence R(V ) of the finite volume density expansion (6.11) by use the results (6.23) and (6.24) of Groeneveld and from the general bound (6.19) will thus find the lower bound for the radius of convergence R of the virial expansion (6.8). We recall that p(z; V ) and ρ(z; V ) are the pressure and density as a function of the fugacity z in finite volume. Thus we may obtain P (ρ; V ) from p(z; V ) and ρ(z; V ) by means of Cauchy’s residue formula as 1 dρ(z; V ) dz P (ρ; V ) = (6.193) p(z; V ) 2πi C dz ρ(z; V ) − ρ where C is a contour in the z plane which surrounds z = 0 and on which the complex number ρ satisfies |ρ| < min |ρ(z; V )|. (6.194) When ρ satisfies (6.194) we may use the expansions of both ∞ 1 ρn = ρ(z; V ) − ρ n=0 ρ(z; V )n+1

(6.195)

½


and P (ρ; V ) =

∞

cn (V )ρn

(6.196)

n=1

in (6.193) to find 1 1 dρ(z; V ) p(z; V ) dz cn (V ) = 2πi C dz ρ(z; V )n+1 1 d −n ρ(z; V ) = dzp(z; V ) − 2πi C ndz and integrate by parts to find 1 cn (V ) = 2πi

C

dp(z; V ) dz . dz n(ρ(z; V ))n

(6.197)

(6.198)

The radius of convergence R(V ) is thus determined by the minimum value of ρ(z; V ) on the contour C R(V ) = minz∈C ρ(z; V ). (6.199) Furthermore by comparison of (6.196) with (6.11) we see that l cl+1 (V ) =− βl (V ) kB T l+1

(6.200)

and thus using the relation (6.6) ρ(z; V ) =

z dp(z; V ) kB T dz

in (6.198) we find 1 lβl (V ) = − 2πi

C

(6.201)

dz . z(ρ(z; V ))l

(6.202)

To study the minimum of ρ(z; V ) we recall that ρ(z; V ) = z +

∞

lbl (V )z l

(6.203)

l=2

where the result (6.24) shows that the series converges if |z| < 1/2eB2. where B2 is the second virial coefficient in infinite volume. Therefore if in (6.203) we use the upper bound (6.23) on bl (V ) we have |ρ(z; V ) − z| = |

∞

lbl (V )z l | ≤

l=2

∞

|lbl (V )||z|l ≤

l=2

∞ 1 l−1 (2B2 |z|)l . (6.204) l 2B2 l! l=2

We now use a lovely formula of Euler which will be proven at the end of this section w=

∞ l−1 l (we−w )l l=1

l!

where the series converges for 0 ≤ w and is unique for 0 ≤ w ≤ 1.

(6.205)


½

Thus, assuming that 2B2 |z| ≤ e−1 , we define w as the smallest positive solution of we−w = 2B2 |z| and write (6.204) as |ρ(z; V ) − z| ≤

(6.206)

w − |z|. 2B2

(6.207)

This upper bound in the distance of ρ(z) from z leads to a lower bound on the distance of ρ(z) from the origin by noting the triangle inequality |ρ(z; V )| ≥ |z| − |ρ(z; V ) − z|

(6.208)

and thus using (6.207) |ρ(z; V )| ≥ |z| −

w w w − |z| = 2|z| − = (2e−w − 1) . 2B2 2B2 2B2

(6.209)

The contour C can be chosen to be any circle with constant |z| such that |z| < 1/e2B2. This is satisfied for w such that 0 < w < 1 and thus we find |ρ(z; V )| ≥ max0≤w≤1 (2e−w − 1) This is maximized when

w . 2B2

(6.210)

1 = e−w 2(1 − w)

(6.211)

wmax = .31492 · · ·

(6.212)

(2e−wmax − 1)wmax = 0.14476 · · ·

(6.213)

from which we find and thus

Thus we obtain from (6.199), (6.210) and (6.213) the desired result R(V ) ≥

0.14476 · · · . 2|B2 |

(6.214)

A bound on |βl (V )| is similarly obtained if in (6.202) we choose the contour C as |z| = constant to obtain 1 1 = (minC |ρ(z; V )|)l . dzmaxC (6.215) l|βl (V )| ≤ 2πi C z(ρ(z; V ))l Therefore, choosing the radius of the circle C as before, we use the bound (6.213) to obtain l 2|B2 | . (6.216) l|βl (V )| ≤ 0.14476 It remains to prove the formula of Euler (6.205) which follows as an application of

½


B¨ urmann’s theorem [12, pp. 128-131] Let φ(z) be an analytic function of z in some neighborhood of zero and let φ (0) = 0. Then for z sufficiently close to zero, any function f (z) analytic about zero may be expanded as f (z) = f (0) +

∞ {φ(z) − φ(0)}m dm−1 [f (z){ψ(z)}m ]z=0 m−1 m! dz m=1

(6.217)

z . φ(z) − φ(0)

(6.218)

where ψ(z) = To prove this we write

1 f (t)φ (ζ) f (ζ)dζ = dζ dt f (z) − f (0) = 2πi C φ(t) − φ(ζ) 0 0 m−1 z ∞ f (t)φ (ζ) φ(ζ) − φ(0) 1 dζ dt = 2πi 0 φ(t) − φ(0) m=1 φ(t) − φ(0) C ∞ z 1 f (t) φ (ζ)(φ(ζ) − φ(0))m−1 = dζ dt . 2πi (φ(t) − φ(0))m C m=1 0

z

z

(6.219)

Then if we use (6.218) we obtain f (z) = f (0) +

∞ m=1 z

= f (0) +

∞ m=1

z

dζφ (ζ)(φ(ζ) − φ(0))m−1

0

dζφ (ζ)(φ(ζ) − φ(0))m−1

0

1 2πi

dt C

f (t)ψ(t)m tm

1 dm−1 (f (t)ψ(t)m ) |t=0 (m − 1)! dtm−1 (6.220)

from which, when we do the integral over ζ, (6.217) follows. The formula (6.205) now follows if in (6.217) we set f (z) = z, φ(z) = ze−z

(6.221)

along with ψ(t) = et ,

6.6

dm−1 mt e |t=0 = mm−1 . dtm−1

(6.222)

Counting of Mayer graphs

We use the following notation: GL(k) is the number of all labeled Mayer graphs (both connected and disconnected) of order k. CL(k) is the number of labeled connected Mayer graphs of order k.

Counting of Mayer graphs

½

IL(k) is the number of labeled irreducible Mayer graphs of order k. GU (k) is the number of all unlabeled Mayer graphs (both connected and disconnected) of order k. CU (k) is the number of unlabeled connected Mayer graphs of order k. IU (k) is the number of unlabeled irreducible Mayer graphs of order k. By direct computation (either by hand or computer assisted) we have (from Table A3 of [13] and Table 1 of [14]) the results listed in Table 6.1. Table 6.1 The number of unlabeled total, connected and irreducible (biconnected) Mayer graphs up to order 14.

k 1 2 3 4 5 6 7 8 9 10 11 12 13 14

GU (k) 1 2 4 11 34 156 1044 12, 346 274, 668 12, 005, 168 1, 018, 997, 864 165, 091, 172, 592 50, 502, 031, 367, 952 29, 054, 155, 657, 235, 488

CU (k) 1 1 2 6 21 112 853 11, 117 261, 080 11, 716, 571 1,006,700,565 164,059,830,476 50, 335, 907, 869, 219 29, 003, 487, 462, 848, 061

IU (k) 0 1 1 3 10 56 468 7, 123 194, 066 9, 743, 542 900, 969, 091 153, 620, 333, 545 48, 432, 939, 150, 704 28, 361, 824, 488, 394, 169

The number of all labeled Mayer graphs of k points is GL(k) = 2k(k−1)/2 .

(6.223)

To prove this we let gl (k) be the number of labeled graphs with k points and l lines. We note that for any set of k points there are k(k − 1)/2 distinct unordered pairs of points. In a graph of l lines these pairs are either connected by a line or they are not and therefore k(k − 1)/2 gl (k) = . (6.224) l Thus we have

k(k−1)/2

k(k−1)/2

GL(k) =

gl (k) =

l=0

=2

l=0 k(k−1)/2

k(k − 1)/2 l

(6.225)

where in the last line we have used the binomial theorem. Thus (6.223) is established.

½


The number of all unlabeled Mayer graphs of k points behaves as k → ∞ as [13, eqn.(9.1.25) p.199]) 2k(k−1)/2 GU (k) = k!

k5 k2 − k (3k − 7)k! + O( 5k/2 ) . 1 + k−1 + 2k 2 2 (3k − 9)(k − 4)! 2

(6.226)

The following results are also proven in [13, chapter 9]. Almost all unlabeled Mayer graphs are connected in the sense that limk→∞

CU (k) = 1. GU (k)

(6.227)

Almost all unlabeled Mayer graphs are irreducible in the sense that limk→∞

IU (k) = 1. GU (k)

(6.228)

The approach to these large k results is apparent in Table 6.1.

6.7

Appendix: The irreducible Mayer graphs of four and five points

We show in Fig. 6.4 the three irreducible unlabeled Mayer graphs of four points and the number of the corresponding labeled graphs. In Fig. 6.5 we show the irreducible Mayer graphs of five points and the number of the corresponding labeled graphs.

Fig. 6.4 The 3 unlabeled irreducible (biconnected) Mayer diagrams with four points. The first number under the graph is the number of labeled graphs corresponding to the unlabeled graph.

Appendix: The irreducible Mayer graphs of four and five points

½

Fig. 6.5 The 10 unlabeled irreducible (biconnected) Mayer diagrams with five points. The first number under the graph is the number of labeled graphs corresponding to the unlabeled graph.

References [1] J.E. Mayer and M.G. Mayer, Statistical Mechanics (Wiley 1940), chapter 13, 277–284. [2] G.E. Uhlenbeck and G.W. Ford, “The theory of linear graphs with applications to the theory of the virial development of the properties of gases” in Studies on Statistical Mechanics, vol. 1, ed. J. de Boer and G.E. Uhlenbeck (North Holland 1962) 123–207. [3] J. Groeneveld, Two theorems on classical many–particle systems, Phys. Letts. 3 (1962) 50–51. [4] J.L. Lebowitz and O. Penrose, Convergence of virial expansions, J. Math. Phys. 5 (1964) 841–847. [5] D. Ruelle, Statistical Mechanics (Benjamin, New York, 1969) [6] E. Lieb, New method in the theory of imperfect gases and liquids, J.Math. Phys. 4 (1963) 671-678. [7] J. Groeneveld, Estimation methods for Mayer graphical expansions, Doctor’s thesis published in Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series 70, Nrs. 4 and 5, 451–507 :(1967c) Part I Graphical Expansions; (1968a) Part II General Procedure; (1968b) Part III Estimation Methods of degrees 0 and 1; (1968c) Part IV Estimation methods of degree 2. [8] D. Ruelle, Correlation functions of classical gases, Ann. Phys. 25 (1963) 109–120. [9] O. Penrose, Convergence of fugacity expansion for fluids and lattice gases, J. Math. Phys. 4 (1963) 1312–1320. [10] J. de Boer, Molecular distribution and the equation of state of gases, Reports on Progress in Physics, XII (1949) 305-374. [11] J. Groeneveld, Estimation methods for Mayer’s graphical expansions, in Graph Theory and Theoretical Physics ed. F. Harary (1967 Academic Press) 229-259. [12] E.T, Whittaker and G.N. Watson, A Course in Modern Analysis, 4th edition (Cambridge University Press 1963). [13] F. Harary and E.M. Palmer, Graphical Enumeration Academic Press (New York and London 1973) [14] R.W. Robinson and T.R. Walsh, Inversion of cyclic index sum relations for 2- and 3- connected graphs, J. Comb. Theory, B57 (1993) 289–308.

7 Ree–Hoover virial expansion and hard particles In this chapter we study what may be considered to be the simplest potential in classical statistical mechanics, hard spheres in D dimensions defined by the potential U (r) =

∞ if |r| ≤ σ 0 if σ ≤ |r|.

(7.1)

Equation (7.1) says that the centers of the two spheres cannot be closer than σ which is traditionally called the hard core diameter of the sphere. The potential (7.1) is particularly simple because the Boltzmann weights are independent of temperature. Thus P/kB T is a function of density alone and the internal energy is the same as the perfect gas. There is no energy scale in the problem and all the physics is determined by entropy, geometry and combinatorics. What we learn from the study of hard spheres is basic to what has become known in recent years as “soft condensed matter physics”. The study of the virial coefficients of the hard sphere gas splits into two separate parts: the generation of the graphs and the evaluation of the integrals. The generation of the Mayer graphs is a serious problem because the number of these graphs grows very rapidly. The computation of integrals is difficult because there are strong cancellations between graphs, some of which are positive and some of which are negative. A considerable simplification in the computation of virial coefficients was made in the mid 1960s by Ree and Hoover [1–3] who introduced a rearrangement of the Mayer graphs which decreases the number of graphs. We present this expansion in section 7.1. For the hard core gas (7.1) the Ree–Hoover expansion has the additional property that when the virial coefficient Bk is evaluated in D dimensions if k − 3 ≥ D then some diagrams vanish for the hard core potential which do not vanish in general. diagrams This vanishing is vividly demonstrated in section 7.2 where we consider the very simple case of the Tonks gas [4], which is the one-dimensional case of the hard sphere gas (7.1), and show that there is exactly one nonvanishing Ree-Hoover graph for each virial coefficient. In section 7.3 we analytically evaluate the virial coefficients for hard spheres B2 , B3 , and B4 [5–13]. The results are given in Tables 7.5 and 7.6. The derivations for B2 and B3 are given in detail. The virial coefficients B5 –B10 have been evaluated numerically [1, 3, 14–17]. We discuss these computations in section 7.4 and tabulate the results in Table 7.8.

½

Ree–Hoover virial expansion and hard particles

In section 7.5 we discuss the possible behavior of the hard sphere virial coefficients for k ≥ 11 and examine the behavior of certain graphs for large values of k and in section 7.6 we use the results to estimate the radius of convergence of the virial series and discuss the location of the leading singularity in the complex density plane. We also present various of the approximate equations of state that have been proposed for hard spheres in three dimensions [18–26]. The hard sphere potential (7.1) is radially symmetric and the spatial variable r is in the continuum. However, the Mayer and Ree–Hoover expansions are equally valid for potentials which have an angular dependence and on a lattice the analogue of (7.1) is the pair potential U (rk − rk ) for a lattice gas with nearest neighbor exclusion U (0) = +∞ U (rj − rk ) = +∞ for rj and rk nearest neighbors on the lattice 0 otherwise

(7.2)

In section 7.7 we present the known results in the continuum for parallel hard squares and cubes [27] and the lattice gas results for hard squares on a square lattice [28, 29], hard hexagons on the triangular lattice [29–33] and hard particles with nearest neighbor exclusion on the cubic, fcc and bcc lattices in three dimensions [30]. In section 7.8 we consider nonspherical convex bodies which are allowed to rotate. In three dimensions the first computation of a second virial coefficient for nonspherical shapes was given by Onsager in 1949 [34], and in two dimensions B2 , B3 and B4 have been computed for ellipses, rectangles and needles [35, 36]. We conclude in section7.9 with a summary of some of the open questions in the study of virial expansions of hard particles.

7.1

The Ree–Hoover expansion

In the previous chapter we showed that the coefficients in the virial expansion of the pressure ∞

Pv Bk v 1−k =1+ kB T

(7.3)

k=2

are Bk = −

k−1 k!

Vk (r1 , · · · , rk )dD r1 · · · dD rk−1

(7.4)

where Vk (r1 , · · · , rk ) is the sum of all numbered biconnected Mayer graphs with k points. In these Mayer graphs each line stands for the factor f (ri,j ) ≡ fi,j = e−U (ri −rj )/kB T − 1.

(7.5)

The numbered Mayer graphs of a certain type are obtained by labeling the vertices of the basic unnumbered graph in all possible distinct ways. Calling si [n] the number


½

of labelings of the graph of type i with n points and calling Si [n] the integral of the unlabeled graph of type i and n points we have Bn = −

n−1 si [n]Si [n]. n! i

(7.6)

In 1964 Ree and Hoover in [1–3] introduced a simple and very useful modification of this expansion by introducing in addition to (7.5) the function f˜ = e−U/kB T

(7.7)

1 = f˜i,j − fi,j .

(7.8)

with the property that The basic idea of the re-expansion is to note that, in a general Mayer graph, points are either connected by a line which contributes a factor of f or they are not connected. Every pair of points which is not connected can be thought of as contributing a factor of 1 to the integral. The Ree–Hoover expansion is to replace this 1 by f˜ − f and to rewrite the expansion in terms of integrals where all points are connected by bonds which now are either f or f˜. As a first example of the utility of this method, consider the fourth virial coefficient. There are three contributing graphs as shown in Fig. 7.1 and the symmetry numbers are s1 [4] = 1, s2 [4] = 6, s3 [4] = 3. (7.9)

Fig. 7.1 The three unlabeled Mayer diagrams which contribute to B4 and their symmetry numbers si [4].

Thus

1 B4 = − {S1 [4] + 6S2 [4] + 3S3 [4]} 8

with

(7.10)

S1 [4] = S2 [4] =

f1,2 f2,3 f3,4 f4,1 f1,3 f2,4 dD r1 · · · dD r3

(7.11)

f1,2 f2,3 f3,4 f4,1 f1,3 dD r1 · · · dD r3

(7.12)

½


f1,2 f2,3 f3,4 f4,1 dD r1 · · · dD r3 .

S3 [4] =

(7.13)

Now in S3 [4] we insert the factors 1 = f˜1,3 − f1,3 , and 1 = f˜2,4 − f2,4

(7.14)

and in S2 [4] we insert the factor 1 = f˜2,4 − f2,4 ,

(7.15)

S1 [4] + 6S2 [4] + 3S3 [4] = −2S˜1 [4] + 3S˜3 [4]

(7.16)

and find that where S˜1 [4] = S˜3 [4] =

f1,2 f2,3 f3,4 f4,1 f1,3 f2,4 dD r1 · · · dD r3

(7.17)

f1,2 f2,3 f3,4 f˜4,1 f1,3 f˜2,4 dD r1 · · · dD r3 .

(7.18)

Hence we have reduced the number of diagrams to be considered from three to two. The computation is summarized graphically in Fig. 7.2 where the dotted lines represent the factor f˜.

Fig. 7.2 The graphical representation of the reduction of the three Mayer diagrams for B4 to the two Ree–Hoover diagrams. The dotted lines represent the factors f˜.

Thus we have the final result B4 =

1˜ 3 S1 [4] − S˜2 [4]. 4 8

(7.19)

To proceed further in a systematic fashion we follow [2] and denote the combinatorial factor in the Ree-Hoover expansion for the Ree–Hoover integral of k points


½

˜i [k]. We refer to a ˜i [k] as the star content of the k point graph of type i. By S˜i [k] as a definition k−1 k−1 si (k)Si (k) = − a ˜i (k)si (k)S˜i (k). (7.20) Bk = − k! k! i i To compute a ˜i [k], consider a Ree–Hoover diagram S˜j (k). This diagram is produced by expanding all those Mayer diagrams whose f functions are a subset of the f functions in S˜j (k). Denote those contributing diagrams as Sl [j, k] and denote by ∆fl the number of f bonds in S˜j (k) that are not in the Mayer diagram Sl [j, k]. It is clear that the Sl [j, k] are exactly those diagrams which can be formed by removing ∆fl f functions from the f functions in S˜j [k]. Therefore we see that a ˜j [k] = (−1)∆f l (7.21) l

where the minus sign appears because the expansion of (f˜ − f ) introduces a minus sign with each of the f functions. The equation (7.21) can be expressed by the following rule [2, p.1637]: Count the number of labeled Mayer graphs which can be formed by successively removing 0, 2, · · · of the f functions from the Ree-Hoover diagram S˜j [k] and subtract from that the number of labeled Mayer graphs that can be formed from removing 1, 3, · · · of the f functions from S˜j [k]. The resulting number (which can be positive, negative or zero) is the star content a ˜j [k]. There is a useful property of the star content a ˜j [k] which helps in the computations. Namely if S˜j [k] and S˜j [k − 1] have the same type of f˜ bonds and differ only in the f bonds then a ˜j [k] = (−1)k−1 (k − 2)˜ aj [k − 1]. (7.22) From this we find recursively that ˜j [m](k − 2)!/(m − 2)! m < k a ˜j [k] = (−1)k(k−1)/2−m(m−1)/2 a

(7.23)

where m is the smallest total number of points possible for the given configuration of f˜ bonds. From (7.23) we find that for the “complete Ree–Hoover star diagram” which contains no bonds f˜ we have m = 2 and a ˜j [2] = 1 and thus we find for the star content astar [k] = −(−1)k(k−1)/2 (k − 2)!

(7.24)

For B5 and B6 we explicitly give all contributing Ree–Hoover graphs in Tables 7.1 and 7.2. Here we use the notation that the diagram specified by Bk [m, i] has k points, and m points are connected by f˜ bonds. We list separately the factors sk [m, i]. a ˜k [m, i] and the product k−1 Ck [m, i] = − sk [m, i]˜ ak [m, i]. (7.25) k! We also specify the graphs in two different notations either by giving the f˜ or the f bonds.

½


Table 7.1 Ree–Hoover diagrams for B5 . For each diagram we give the values of the combina˜k [m, i] and the product Ck [m, i] = − k−1 sk [m, i]˜ ak [m, i] torial factor sk [m, i], the star content a k! where m is the number of points conected by f˜ bonds.

Label

sk [m, i]

a ˜k [m, i]

Ck [m, i]

B5 [0, 1]

1

−6

6/30

B5 [4, 1]

15

3

−45/30

B5 [5, 1]

30

−2

60/30

B5 [5, 2]

12

1

−12/30

B5 [5, 3]

10

1

−10/30

f˜ notation

f notation

∅

The Ree–Hoover expansion has two definite advantages over the Mayer expansion. The first is that for any potential the number of Ree–Hoover graphs is smaller than the number of Mayer graphs. The second advantage is that for low dimensions certain diagrams vanish because of geometrical constraints. This effect is seen by examining B5 [5, 3] where in D = 2 (but not for D ≥ 3) it is impossible to find any configuration which satisfies both the restrictions that f (r) = 0 for |r| ≥ σ and f˜(r) = 0 for |r| < σ. The number of contributing diagrams for general potentials, hard disks and hard spheres is given in Table 7.3 up through B10 . The values of these virial coefficients will be computed in section 7.3 and 7.4.

7.2

The Tonks Gas

The Tonks gas [4] is the name given to the particularly simple case of the hard sphere gas (7.1) in one dimension. In this case the partition function is particularly easy to compute if we note that a collision of hard rods in one dimension in a volume L behaves kinematically in the same way as free particles which move in volume of L − N a where N is the number of rods. Thus if we replace V by L − N a in the equation of state of the free gas we have the equation of state of the Tonks gas P (v − σ) = 1. kB T

(7.26)

When this is rewritten in the form of the virial expansion (7.3) we find ∞

1 Pv = =1+ (σ/v)k kB T 1 − σ/v

(7.27)

Bk = σ k−1 = B2k−1

(7.28)

k=1

and thus which is to be compared with the bound (6.40) |Bk |/B2k−1 ≤

0.21780 · · · (7.1823 · · ·)k−1 k

(7.29)

The Tonks Gas

½

Table 7.2 Ree–Hoover diagrams for B6 . For each diagram we give the values of the combina˜k [m, i] and the product Ck [m, i] = − k−1 sk [m, i]˜ ak [m, i] torial factor sk [m, i], the star content a k! where M is the nmber of points connected by f˜ bonds.

sk [m, i]

a ˜k [m, i]

Ck [m, i]

f˜ notation

B6 [0, 1]

1

24

−24/144

∅

B6 [4, 1]

45

−12

540/144

B6 [5, 1]

180

8

−1440/144

B6 [5, 2]

72

−4

288/144

B6 [5, 3]

60

−4

240/144

B6 [6, 1]

360

3

−1080/144

B6 [6, 2]

180

−2

360/144

B6 [6, 3]

60

1

−60/144

B6 [6, 4]

60

−6

360/144

B6 [6, 5]

180

−5

900/144

B6 [6, 6]

90

−4

360/144

B6 [6, 7]

45

4

−180/144

B6 [6, 8]

360

−1

360/144

B6 [6, 9]

360

−2

720/144

B6 [6, 10]

60

4

−240/144

B6 [6, 11]

15

16

−240/144

B6 [6, 12]

180

3

−540/144

B6 [6.13]

360

1

360/144

B6 [6, 14]

90

−2

180/144

B6 [6, 15]

90

−1

90/144

B6 [6, 16]

180

1

−180/144

B6 [6, 17]

15

1

−15/144

B6 [6, 18]

10

4

−40/144

Label

f notation

½


Table 7.3 The number of Mayer and Ree–Hoover graphs which contribute to the virial coefficients up to order 10 from [17]. Some of the entries are only lower bounds because it is numerically difficult at times to distinguish between graphs which are very small and those which vanish identically.

B2 B3 B4 B5 B6 B7 B8 B9 B10

Mayer 1 1 3 10 56 468 7, 123 194, 066 9, 743, 542

R–H in general 1 1 2 5 23 171 2, 606 81,564 4,980,756

R–H,D = 2 1 1 2 4 15 73 > 647 ∼ > ∼ 8,417 > ∼ 110,529

R–H,D = 3 1 1 2 5 22 161 > 2334 > 60, 902

R–H,D = 4 1 1 2 5 23 169 > 2556 > 76, 318

The Ree–Hoover expansion provides an exceptionally simple derivation of (7.27) because it is easily seen that in the evaluation of Bk all graphs vanish except the one in which each vertex is connected with every other vertex by the bond fi,j . Therefore we find from (7.20) and (7.24) (−1)k(k−1)/2 dx2 · · · dxk Bk = fi,j k 1≤i<j≤k 1 dx2 · · · dxk = |fi,j |. (7.30) k 1≤i<j≤k

where, to obtain the last line, we have used the fact that the product contains k(k−1)/2 terms which are either −1 or zero. If we order the coordinates xj as x1 < x2 < · · · < xk and set x1 = 0 by convention we see that (7.30) consists of k! integration regions all of which contribute equally. Thus we obtain σ σ σ 1 Bk = k! dx2 dx3 · · · dxk k 0 x2 xk−1 σ σ σ 1 1 = k! dx2 dx3 · · · dxk = σ k−1 (7.31) k (k − 1)! 0 0 0 which agrees with (7.28). It is also instructive to obtain the cluster integrals bk . We first use (6.128) to find ∞ k+1 k k σ ρ . (7.32) z = ρ exp k k=1

Then noting that ∞ ∞ k+1 k k 1 k k σρ σρ σ ρ = + σ ρ = − ln(1 − σρ) k 1 − σρ k 1 − σρ k=1

k=1

(7.33)

Hard sphere virial coefficients B2 –B4 in two and higher dimensions

½

we have

P σP exp . kB T kB T We now use B¨ urmann’s theorem (6.217) with P as the variable z and set z=

f (P ) = P, φ(P ) =

σP P exp = z, kB T kB T

(7.34)

(7.35)

where ψ(P ) = kB T exp(−

dm−1 σP ), ψ(P )m |P =0 = kB T (−mσ)m−1 kB T dP m−1

to obtain

∞ zm P = (−mσ)m−1 . kB T m! m=1

(7.36)

(7.37)

Thus comparing with (6.5) we find bk =

1 (−kσ)k−1 , k!

(7.38)

from which we find b2 = −σ and thus bk (k/2)k−1 = k−1 (2b2 ) k!

(7.39)

which should be compared with Groeneveld’s bounds (6.23).

7.3


We now turn to the evaluation of Bk for dimensions D ≥ 2. The chronology of the computations is given in Table 7.4 The results for B2 , B3 and B4 are given in Tables 7.5 and 7.6. We will here derive the results for B2 and B3 . The derivation of the results for B4 is somewhat tedious and we refer the reader to the original papers. 7.3.1

Evaluation of B2

The second virial coefficient for the hard sphere potential (7.1) is 1 f (r)dD r B2 = − 2

(7.40)

where f (r) = −1 for |r| ≤ σ 0 otherwise

(7.41)

and therefore

1 VD (σ) 2 where VD (σ) is the volume of a sphere of radius σ in dimension D. B2 =

(7.42)

½


Table 7.4 Chronology of analytic computations of the hard sphere virial coefficients B3 and B4 .

Date 1899 1899 1936 1951 1964 1982 2003 2005

Author(s) Boltzmann [5] Boltzmann [5],van Laar [6] Tonks [4] Nijboer, van Hove [7] Rowlinson [8], Hemmer [9] Luban, Barum [10] Clisby, McCoy [11] Lyberg [12]

Property B3 for D = 3 B4 for D = 3 B3 for D = 2 Two center evaluation of B4 in D = 3 B4 for D = 2 B3 for arbitrary D B4 for D = 4, 6, 8, 10, 12 B4 for D = 5, 7, 9, 11

One way to evaluate VD (r) is to note that (from dimensional considerations) VD (r) = CD rD

(7.43)

and that furthermore if we denote the surface area of the D-dimensional hypersphere by ΩD−1 rD−1 that dVD (r) = DCD rD−1 . (7.44) ΩD−1 rD−1 = dr To evaluate CD we consider the integral

D ∞ ∞ ∞ −(x21 +···+x2D ) −x21 dx1 · · · dxD e = dx1 e = π D/2 . (7.45) −∞

−∞

−∞

This integral is also expressed in terms of CD as using dD r = ΩD−1 rD−1 dr as

∞ ∞ 2 2 2 dx1 · · · dxD e−(x1 +···+xD ) = drΩD−1 rD−1 e−r −∞ −∞ 0 ∞ ∞ D 1 D D−1 −r 2 = DCD drr e = DCD dtt 2 −1 e−t = CD Γ( + 1). 2 2 0 0

(7.46)

∞

Therefore CD =

π D/2 Γ( D 2 + 1)

and thus we have VD (r) =

(7.47)

(7.48)

π D/2 rD Γ( D 2 + 1)

(7.49)

2π D/2 . Γ( D 2)

(7.50)

and ΩD−1 =


Thus in any dimension D we have the desired answer 2N N σ π /(2N !) if D = 2N ΩD−1 aD σ D π D/2 2N +1 N +1/2 = σ B2 = = π D if D = 2N + 1. N 2D 2Γ( + 1) 2

2

l=0

½

(7.51)

(l+1/2)

For large D we use Stirling’s formula that for z → ∞

to obtain

Γ(z + a) ∼ (2π)1/2 e(z+a−1/2) ln z−z

(7.52)

B2 ∼ σ D π (D−1)/2 2−3/2 e1+D/2 e− 2 (D+1) ln(1+D/2) .

(7.53)

1

For low dimensions B2 is explicitly given in Table 7.5. Table 7.5 Exact and decimal results for B2 and B3 for 2 ≤ D ≤ 12

D 2 3 4 5 6 7 8 9 10 11 12

7.3.2

B2 πσ 2 /2 2πσ 3 /3 π 2 σ 4 /4 4π 2 σ 5 /15 π 3 σ 6 /12 8π 2 σ 7 /105 π 4 σ 8 /48 16π 4 σ 9 /48 π 5 σ 10 /240 32π 5 σ 11 /10395 π 6 σ 12 /1440

B3 /B√22 4 3 3 − π 5/8 √ 4 33 3 − π 2 7 53/2√ 4 39 3 − π 5 10 289/2 √ 4 3 279 − 3 π 140 15 6343/2 √ 4 3 297 3 − π 140 18 35995/2 √ 4 3 243 3 − π 110

B3 /B22 decimal 0.78200443 · · · 0.625 0.50633990 · · · 0.4140625 0.34094132 · · · 0.28222656 · · · 0.23461360 · · · 0.19357299 · · · 0.16372846 · · · 0.13731002 · · · 0.11539768 · · ·

Evaluation of B3

The third virial coefficient is given by 1 dD r1 dD r2 f (r1,2 )f (r2 )f (r1 ) 3 1 D d r1 f (r1 ) dD r2 f (r1,2 )f (r2 ) =− 3

B3 = −

Since f (r) is −1 for |r| ≤ a and 0 for |r| > a, 1 B3 = dD r1 dD r2 f (r1,2 )f (r2 ). 3 |r1 | 0 in the high density phase, Ψ = 0 in the low density phase and in the transition region the low and high density phases are connected by a tie line just as for hard spheres. However, a second alternative exists [19–22] because in a disordered region where Ψ = 0 there are two possible behaviors of g6 (r): for r → ∞ either the correlation g6 (r) decays exponentially g6 (r) ∼ e−r/ξ (8.35) or it decays algebraically

g6 (r) ∼ r−η .

(8.36)

We thus have a scenario where for ρ < ρf there is exponential decay of g6 (r) while for ρf < ρ < ρs the correlation g6 (r) decays algebraically. Such a phase, if it exists, is called the hexatic phase and in this region of density the pressure will not be constant but will be monotonic increasing. Since the inception of this hexatic phase there has been much effort to determine which of the two possible scenarios is correct for hard discs. To date there is no definitive answer to the question. As an example we quote the conclusion of the 2002 paper of Binder, Sengupta and Nielaba [23]: It has been shown that the currently available simulation data are compatible with a continuous transition from the fluid to the hexatic phase (with divergent bond orientation susceptibility) at ρf = 0.899 and with a hexatic to crystal transition at ρ ∼ 0.914 ± 0.002. However, no simulations that reach full thermal equilibrium in the density range 0.90 ≤ ρ ≤ 0.915 and show directly the existence of the hexatic phase are available so far. Without such direct evidence, the possibility of a (very weak) first order transition from the fluid to the crystal cannot yet be firmly ruled out, although so far clear signals of two phase coexistence are also lacking.

8.3

The inverse power law potential

The next simplest potential after the hard sphere potential is the inverse power law potential (8.1) which is the only potential that has the property that the free energy depends on one combination of the variable T and ρ and does not depend on T and ρ separately. Thus, while first order freezing transitions are allowed, triple points are forbidden. In the limit that n → ∞ the inverse power law potential (8.1) reduces to

The inverse power law potential

¾¾¿

the hard sphere potential and thus it is expected that for sufficiently large n there will be a freezing transition to an fcc solid. We will see, however, that this phase diagram is valid only for n > 7. For 3 ≤ n ≤ 7 the system develops a second phase transition with a bcc phase lying between the fluid and the fcc phase. We first present the scaling argument which reduces the thermodynamic functions to depend on one variable rather than T and ρ separately and will then discuss the numerical computations of the phase diagram and the evidence for a second bcc phase. 8.3.1

Scaling behavior

For the power law potential (8.1) the Mayer function is σ fij (ri − rj ) = exp − ( )n − 1 kB T |ri − rj |

(8.37)

and therefore every integral in the Mayer or the Ree-Hoover expansion for the virial coefficient Bk is of the form n

σ D D Ik (T ) = d r2 · · · d rk Fk . (8.38) kB T |ri − rj | If we now make the substitution rj = (kB T /)− n σxj 1

(8.39)

we find Ik (T ) = (kB T /)−D(k−1)/n σ D(k−1) and therefore

Bk (T ) =

dD x2 · · · dD xk Fk Dn (k−1)

kB T

1 |xi − xj |

˜k σ D(k−1) B

n

(8.40)

(8.41)

˜k is independent of T, and σ. Thus we have where B ∞

Pv ˜k σ D(k−1) =1+ B kB T

k=2

Therefore if we define

ρ˜ = σ D

D(k−1)/n

ρk−1 .

kB T D/n ρ

kB T

we see that 1+D/n

P = (kB T )

(8.42)

(σ1/n )−D

ρ˜ +

(8.43) ∞

˜k ρ˜k B

.

(8.44)

k=2

This is valid in the low density regime where the virial expansion is valid. However, an identical argument can be made on the partition function itself which is valid for

¾¾

High density expansions

all densities. Therefore we conclude for the potential (8.1) that P/(kB T )1+D/n is a function of the single variable v(kB T )D/n and is not a function of v and T separately. In the limit of hard spheres this reduces to the statement that P/kB T is a function only of the density ρ and is independent of the temperature. The repulsive power law potential (8.1) is the only potential where the equation of state effectively depends on only one instead of two variables. For hard spheres and power law potentials the special feature that P/(kB T )1+D/n depends only on the single variable v(kT )D/n means that if there is a flat portion for one isotherm then there will be a flat portion on all isotherms. In Fig. 8.7 we schematically plot the (P, v) diagram for a power law potential with some finite value of n for both the case of one and two phase transitions.

fcc

fcc P

fluid

v

P

bcc fluid

v

Fig. 8.7 A schematic plot of isotherms for a power law potential with some finite value of n. The plot on the left has single fcc crystalline. The plot on the right has both an fcc and a bcc phase.The endpoints of the phases are proportional to (kB T )−3/n .

8.3.2

Numerical computations

The molecular dynamics computations for inverse power law potentials are substantially more involved than for hard spheres for several reasons. We will only sketch here the highlights of the procedure. A full explanation is to be found in the original papers [24–28]. First of all there is now no well-defined notion of collision, and the equations of motion for all N particles have to be integrated for (small) discrete time steps. Secondly it must be realized that there is no temperature variable per se in a molecular dynamics computation. In these dynamical computations it is the energy which is kept constant and not the temperature (which does not appear). The temperature is kept constant by rescaling the velocities after every time step. However, because of the scaling property of this potential only one isotherm needs to be computed. With these two modifications the calculations in the fluid phase may now be done in a manner identical to the corresponding fluid phase computations done for hard spheres. The calculations in the bcc and fcc phases are more difficult than for hard spheres for two reasons. Firstly there is the need to keep the bcc and fcc phases in their proper ergodic component, and secondly the need to find a method of computing the

Hard spheres with an additional square well

¾¾

integration constant in the free energy for the (P, v) data computed by molecular dynamics which is compatible with the integration constant in the fluid phase free energy. For example in [27, page 75] the following procedure is used: Here the system is transformed continuously through the use of a coupling constant denoted here by λ, from the fully interacting N particle system (λ = 0) to a collection of N independent (Einstein) harmonic oscillators centered at the lattice sites of the crystal, for which the free energy can be calculated analytically. The coupling to the lattice sites prevents the system from melting as the interparticle pair potentials are scaled to zero. Note that we have assumed that the crystal has zero concentration of vacancies, since density functional methods show that the small equilibrium concentration of vacancies makes a negligible contribution to the bulk free energy. The lattice coupling potential energy used in the simulation of [27] is   Φ(λ) = Φ0 + (1 − λ)2  (r0 /|rk − rj |)n − Φ0  + λkmax (Ri − R0i )2 (8.45) k<j

i

where Φ0 is the (N = ∞) static lattice energy, {R0i } is the set of crystal lattice sites, and kmax is chosen such that the mean squared lattice displacement of the Einstein oscillators is approximately that of the actual uncoupled system. Furthermore [27, page 76] has the proviso Since the system becomes increasingly non-ergodic as the Einstein crystal limit λ → 1 is approached, the molecular dynamics procedure was modified so as to reset periodically the particle velocities from a Boltzmann distribution defined by the temperature, allowing a more complete sampling of the phase space. Finally once the free energies for the separate phases are computed the tie line between the phases is computed using a double tangent construction. There are thus several assumptions made in these computations which are not present in the previous computations for hard spheres. Over the years there have been a variety of numerical simulations of the inverse power law potential: the fluid and fcc phases for n = 12 in [24]; the fluid and fcc phases for n = 4, 6, 9 in [25]; the fluid, bcc and fcc phases for n = 4, 6 in [26]; the fluid, bcc and fcc phases for n = 6 in [27]; the fluid, bcc and fcc phases for n = 4, 6, 9, 12 in [28]. Each paper uses somewhat different assumptions and numerical methods but even though there are some quantitative differences in the various results they all confirm the qualitative three-phase picture of Fig. 8.7 for 3 ≤ n ≤ 7.

8.4

Hard spheres with an additional square well

The next potential to consider is the square well potential (8.4) which may be either repulsive or attractive depending on the sign in front of . Molecular dynamics studies of these potentials were initiated by Young and Alder for the repulsive step potential [29] in 1979 and for the attractive square well [30] in 1980. In contrast to the hard sphere and inverse power law potentials these potentials depend on density and temperature separately. It is thus possible to obtain triple points and critical points in addition to lines of first order transitions.

¾¾


We restrict ourselves here to the case of the attractive square well (8.4) with the negative sign, which has become an important model for the study of real liquids. We present here the results of three selected studies of this system [30–32] which reveal a rich variety of phenomena that change as the width of the attractive region is varied. As in the case of the potentials previously studied the computations have various approximations and caveats which must be taken into consideration when evaluating these results. The pioneering paper of [30] made a molecular dynamics study of the particular case c = 1.5. The results of this study are reproduced in Figs. 8.8 and 8.9 .These results have the very striking feature that, in addition to the expected liquid–vapor critical point and the liquid–vapor–solid triple point, there is a first order transition to a hex close packed phase and, even more remarkable, there is an isostructural transition in the fcc phase that ends in a critical point.

25

20 FCC HCP 80

FCC

15 P*

A FCC

60

P*

10

B

HCP

40

C FCC

Liquid

5

20 Liquid 0

0

1

2 T*

3

0

D 0

E 0.5

1.0 T*

1.5

Fig. 8.8 The phase diagram of the attractive square well potential with c = 1.5 taken from [30]. The figure on the right is an enlarged detail of the figure on the left where the labeled points are (A) the fcc–fcc critical point; (B) hcp–fcc–fcc triple point; (C) hcp–fcc–liquid triple point; (D) hcp-liquid–vapor triple point;(E) liquid–vapor critical point. In both figures the horizontal axis is T ∗ = kB T / and the vertical axis is P ∗ = P/ρcp .

Further study of this isostructural transition was made in [31] by varying the parameter c in the range 1.01 ≤ c ≤ 1.06 and it was found that as c in increased the distance between the hcp–fcc–fcc triple point and the fcc–fcc critical point decreases. Quantitatively, however, the fact that the isostructural transition disappeared at c = 1.06 is at variance with the observation of the transition in [30] at c = 1.5. Finally a more recent numerical study in conjunction with additional theoretical considerations [32] has reported a much more elaborate structure of phases than re-

Lennard-Jones potentials

¾¾

2.5 FCC

2.0

FCC-Fluid

FCC FCC-FCC

HCP

Fluid

T*

1.5

HCP-FCC

1.0

HCP-Fluid Liquid-Vapor

0.5

HCP-Vapor 0 1.0

1.2

1.4 V*

1.6

1.8

Fig. 8.9 Phase boundaries of the c = 1.5 square-well potential in the T –V plane from [30]. The horizontal axis is v/vcp and the vertical axis is T ∗ = kB T /.

ported in either [30] or [31] which we reproduce in Fig. 8.10 where the numbers in parenthesis indicate the number of atoms in a unit cell.

8.5

Lennard-Jones potentials

The final potential we discuss is the famous Lennard-Jones (6, 12) potential (8.5) which was first introduced in 1924 [33] and has been used many times to model the behavior of the noble gases which have the simple phase diagrams given in chapter 2 with one solid fcc phase, an fcc–fluid–vapor triple point and a fluid–vapor critical point. Many numerical computations have been done on this system beginning with the pioneering computation in 1969 of Hansen and Verlet [35] which yielded a phase diagram in qualitative agreement with the noble gases. However, instead of being an example of the successes of numerical computations this agreement of numerical computations with experiment is actually, in the words of Choi, Ree and Ree [36] a “continuing scandal” because it has been known ever since the work of Kihara and Koba [34] in 1952 that, at low temperature and low pressure, the crystalline structure of the Lennard-Jones potential is not the face centered cubic lattice with one atom per unit cell observed in the noble gases but is rather the hex close packed lattice with two atoms per unit cell. As shown by Stillinger [37] an fcc phase at T = 0 can only be produced by putting the system under sufficiently high pressure. The phase boundary between the fcc and hcp phases has been studied in some detail in [36] and the phase diagram looks qualitatively like the phase diagram

¾¾


Pσ3/ε

70 50 30 25

fcc(12)

fcc(18)

20

0 0.0

ct(14)

4)

1 ct’(

10 5

)

14

ct(

15

bcc(14)

hex(20)

0.5

liq

1.0

1.5

kBT/ε Fig. 8.10 Phase boundaries of the attractive square well potential for c = 1.43 as given in [32]

of the attractive square well of Fig. 8.8 with the exception that for the Lennard-Jones potential no isostructural transition in the fcc phase has been observed. Because of the disagreement of many numerical simulations with the results of Kihara and Koba it is perhaps appropriate to conclude with some discussion of how such discrepancies can arise in addition to the caveats already discussed. First of all there is the almost irresistible tendency to perform simulations with an hcp incompatible cubic box with periodic boundary conditions, and then to choose an integer number N of particles which is a “magic number” for the fcc structure. This alone will strongly predispose towards the formation of fcc versus hcp in finite a N simulation. Furthermore some simulations will cut off the Lennard-Jones potential at some finite distance for numerical simplicity and thus the potential is actually not Lennard-Jones at all. It is possible that such a truncated potential could have an fcc structure and low temperature and pressure but such a potential is in fact not Lennard-Jones.

8.6

Conclusions

Statistical mechanics as presented in chapter 1 of this book holds out the promise of explaining the macroscopic properties of bulk matter in equilibrium from the knowledge of the underlying microscopic interactions. One of the most fundamental properties of a macroscopic system that we would like to know is the phase diagram, and many such examples of phase diagrams were given in chapter 2. The most prominent feature of these diagrams is that there are lines of first order transitions which separate phases with different order parameters. It is these first order transitions that we have investigated with the model potentials in this chapter and because of the ubiquity of these transitions it is disappointing that even for these simple potentials there are so

Conclusions

¾¾

few exact results known and that there are so many unresolved controversies in the numerical results. To quote again from [36] there is a general lack of reliable theoretical techniques to predict the theoretical structure of the simplest crystalline solids from a knowledge of their chemical composition. If we do not even have a satisfactory explanation of why the heavy noble gas solids exist in an fcc rather than a hcp phase we clearly have much to learn about how to use the machinery of statistical mechanics to actually predict the properties of bulk matter.

References [1] B.J. Alder and T.E. Wainwright, Studies in molecular dynamics I. General method, J. Chem. Phys. 31 (1959) 459–466. [2] B.J. Alder and T.E Wainwright, Studies in molecular dynamics II: Behavior of small numbers of elastic hard spheres, J. Chem. Phys. 33 (1960) 1439–1451. [3] W.G. Hoover and B.J. Alder, Studies in molecular dynamics IV. The pressure, collision rate and their number dependence for hard disks, J. Chem. Phys. 46 (1967) 686–691. [4] B.J. Alder, W.G. Hoover and D.A. Young, Studies in molecular dynamics. V. High-density equation of state and entropy for hard discs and sphere, J. Chem. Phys. 49 (1968) 3688–3696. [5] T.L. Hill, Statistical Mechanics (McGraw-Hill, NY 1956) [6] M.E. Fisher, Bounds for the derivatives of the free energy and pressure of a hardcore system near close packing, J. Chem. Phys. 42 (1965) 3852–3856. [7] B.J. Alder and T.E. Wainwright, Phase transition for a hard sphere system, J. Chem. Phys. 27 (1957) 1208–1209. [8] W.W. Wood and J.D. Jacobson, Preliminary results from a recalculation of the Monte-Carlo equation of state of hard spheres, J. Chem. Phys. 27 (1957) 1207– 1208, [9] J.R. Erpenback and W.W. Wood, Molecular dynamics calculations of the hardsphere equation of state, J. Stat. Phys. 35 (1984) 321–340. [10] R.J. Speedy, Pressure of the metastable hard-sphere fluid, J. Phys.: Cond. Matt. 9 (1997) 8591–8599. [11] M.E. Fisher, Theory of condensation and the critical point, Physics 3 (1967) 255– 283. [12] J.S. Langer, Theory of the condensation point, Ann. Phys. 41 (1967) 108–157. [13] W. G. Hoover and F.H. Ree, Melting transition and communal entropy for hard spheres, J. Chem. Phys. 49 (1968) 3609–3617 [14] B.J. Alder and T.E. Wainwright, Phase transitions in elastic disks, Phys. Rev. 127 (1962) 359–361. [15] J.A. Zollweg, G.V. Chester and P.W. Leung, Size dependent properties of twodimensional solids, Phys. Rev. B39 (1989) 9518–9530. [16] J.A. Zollweg and G.V. Chester, Melting in two dimensions, Phys. Rev. 46 (1992) 11186–11189. [17] J. Lee and K.J. Strandburg, First-order melting transition of the hard disk system, Phys. Rev. B46 (1992) 11190–11193. [18] A. Jaster, Computer simulations of the two–dimensional melting transition using hard discs, Phys. Rev. E 59 (1999) 2594–2602.

References

¾¿½

[19] J. M. Kosterlitz and D.J. Thouless, Ordering, metastability and phase transitions in two dimensional systems, J. Phys. C6 (1973) 1181–1203. [20] B.I. Halperin and D.R. Nelson, Theory of two-dimensional melting, Phys. Rev. Lett. 41 (1978) 121–124. [21] D.R. Nelson and B.I. Halperin, Dislocation mediated melting in 2 dimensions, Phys. Rev. B 19 (1979) 2457–2484. [22] A.P. Young, Melting and the vector Coulomb gas in 2 dimensions Phys. Rev. B 19 (1979) 1855–1866. [23] K. Binder, S. Sangupta and P. Nielaba, The liquid–solid transition of hard discs: first order transition or Kosterlitz–Thouless–Halperin–Nelson–Young scenario. J. Phys. Cond. Matt. 14 (2002) 2323–2333. [24] W.G. Hoover, M. Ross, K.W. Johnson, D. Henderson, J.A. Barker and B.C. Brown, Soft-sphere equation of state, J. Chem. Phys. 52 (1970) 4931–4941. [25] W.G. Hoover, S.G. Gray, and K.W. Johnson, Thermodynamic properties of the fluid and solid phases for inverse power potentials, J.Chem. Phys. 55 (1971) 1128– 1136. [26] W.G. Hoover, D.A. Young and R. Grover, Statistical mechanics of phase diagrams. I. Inverse power potentials and the close-packed to body centered cubic transition, J. Chem. Phys. 56 (1972) 2207–2210. [27] B.B. Laird and A.D.J. Haymet, Phase diagram for the inverse sixth power potential system from molecular dynamics computer simulation, Mol. Phys. 75 (1992) 71–80. [28] R. Agrawal and D.A. Kofke, Solid–fluid coexistence for inverse-power potentials, Phys. Rev. letts. 74 (1995) 122–125. [29] D.A. Young and B.J. Alder, Studies in molecular dynamics XVII. Phase diagrams for “step” potentials in two and three dimensions, J. Chem. Phys. 70 (1979) 473– 481. [30] D.A. Young and B. J. Alder, Studies in molecular dynamics XVIII, Square well phase diagram, J.Chem Phys. 73 (1980) 2430–2434. [31] P. Bolhuis, M. Hagen, D. Frenkel, Isostructural solid–solid transition in crystalline systems with short-ranged interactions, Phys. Rev. E 50 (1994) 4880–4890. [32] J. Serrano-Illan, G. Navascues and E. Velasco, Noncompact crystalline solids in the square-well potential, Phys. Rev. E73 (2006) 01110–(1–11). [33] J.E. Lennard-Jones, Proc. R. Soc. London, ser. A 106 (1924),463 [34] T. Kihara and S. Koba, Crystal structures and intermolecular forces of rare gases, J. Phys. Soc. Jpn. 7 (1952) 348–354. [35] J-P Hansen and L. Verlet, Phase transitions of the Lennard-Jones system, Phys. Rev. 184 (1969) 151–161. [36] Y. Choi, T. Ree and F.H. Ree, Phase diagram of a Lennard-Jones solid, J. Chem. Phys. 99 (1993) 9917–9919. [37] F.R. Stillinger, Lattice sums and their phase diagram implications for the classical Lennard-Jones model, J. Chem. Phys. 115 (2001) 5208–5212.

9 High temperature expansions for magnets at H = 0 In this chapter we will develop the theory of high temperature series expansions for the classical n vector and the quantum spin S Heisenberg model . Our goal is to estimate the critical exponents α for the specific heat and γ for the zero field susceptibility introduced in the general theory of critical phenomena and to test the conjecture of universality presented in chapter 5. We write the Hamiltonian for the classical n vector model on a lattice of N sites as H = −Jn nR · nR − H nzR − Hs ηR nzR (9.1) R,R

R

R

n2R

= 1. The sum over R and R is over nearest neighbor pairs (counted once) where of an appropriate lattice. The term with ηR is to be included for the square, cubic and bcc (bipartite) lattices and ηR = ±1 on the respective sublattices. When J > 0 the model is ferromagnetic, and H couples to the magnetization M (H, T ) =

1 z nR . N

(9.2)

R

When J < 0 the model is antiferromagnetic and on the cubic and bcc (bipartite) lattices Hs couples to the staggered magnetization Ms (Hs , T ) =

1 ηR nzR . N

(9.3)

R

We will concentrate on the particular cases of n = 1, 2, 3 which are called Ising, XY and Heisenberg models respectively. We write the Hamiltonian of the spin S quantum Heisenberg model as z z ˜ ˜s H = −J˜ SR · SR − H SR −H ηR SR (9.4) R,R

R

R

where, again, the sum is over nearest neighbor pairs counted once, the quantum spin operators satisfy the commutation relations y y y x z z x z x [SR , SR [SR , SR [SR , SR ] = iSR δR,R , ] = iS δR,R ] = iSR δR,R , R

and

(9.5)

High temperature expansions for magnets at H = 0

S2 = S(S + 1).

¾¿¿

(9.6)

The classical n vector model with n = 3 is the limit S → ∞ of the spin S Heisenberg model if we set ˜ ˜ ˜ s [S(S + 1)]1/2 = Hs JS(S + 1) = 3J, H[S(S + 1)]1/2 = H, H and

SR . S→∞ [S(S + 1)]1/2

nR = lim

(9.7)

(9.8)

We will thus normalize the magnetizations for the quantum case as M (H, T ) =

z 1 SR , N [S(S + 1)]1/2 R

Ms (Hs , T ) =

z 1 SR ηR N [S(S + 1)]1/2 R

(9.9)

and define kB T χ(T ) =

∂M (T, H) ∂Ms (T, Hs ) |H=0 , kB T χs (T ) = |Hs =0 . ∂H ∂Hs

(9.10)

The Hamiltonians for the classical n vector model on the bipartite lattices is invariant if J → −J and nR → −nR on one of the sublattices. Therefore for the n vector model on a bipartite lattice the specific heat at H = Hs = 0 has the symmetry in J/kB T c(−J/kB T ) = c(J/kB T ) (9.11) and the two susceptibilities satisfy χs (−J/kB T ) = χ(J/kB T )

(9.12)

and thus the ferromagnetic and antiferromagnetic cases are essentially identical. For the quantum case the symmetries (9.11) and (9.12) do not hold and we will need to study the ferromagnetic and antiferromagnetic cases separately by giving series both for χ(T ) and χs (T ). High temperature series expansions for these lattice models were developed in the 1950s and 1960s. They are all combinatorial graph enumeration problems and do not involve any integrations as were needed for virial coefficients of the continuum fluids. We study the n vector model in section 9.1. We restrict our attention to the expansions for D = 2 and D = 3 where the expansions have been carried out to high order. From these high-order expansions we see that in D = 2 there is a striking difference between n ≤ 2 where all known coefficients of the susceptibility are positive and hence the leading singularity is on the real axis, and n > 2 where the evidence is that negative coefficients occur and that the leading singularity is in the complex plane. We estimate these radii of convergence first by means of a ratio analysis and then by the methods of Padé and differential approximates. Wherever possible we use these series to estimate the critical exponents γ and α. The spin S quantum Heisenberg model is studied in section 9.2. The series for these quantum models are shorter than for the classical n vector model. We investigate in

¾


detail the evidence for the prediction of universality that the critical exponents for both the ferromagnetic and antiferromagnetic models will be independent of the quantum spin S. In section 9.3 we discuss the significance of these computations for magnets and conclude in section 9.4 with the interpretation of the classical n vector model as a quantum field theory and in particular how the classical n = 3 Heisenberg magnet in two dimensions illustrates the concept of asymptotic freedom.

9.1

Classical n vector model for D = 2, 3

Partition functions for continuum fluids interacting via two-body potentials U (r) are integrals over the Boltzmann weights e−U (r)/kB T

(9.13)

and if the two-body potential U (r) were finite for all r then the limit T → ∞ would be trivial and an expansion about T = ∞ could be developed. Unfortunately as we have seen in previous chapters all realistic continuum potentials are infinite at r = 0 and hence expansions about T → ∞ are not possible. This was illustrated most vividly for the hard sphere gas where there is only one isotherm. However the Hamiltonian for the classical n vector model (9.1) does not contain any infinite terms and thus in the partition function Z= dnR e−H/kB T (9.14) R

the limit T → ∞ is trivially given by Z = SnN

(9.15)

where N is the number of sites of the lattice and Sn is the surface area of the ndimensional sphere. Thus we may obtain a high temperature series expansion by expanding the exponential in (9.14) as Z = SnN

∞ j=0

with µj =

1 SnN

µn (kB T )j j!

dnR (−H)j .

(9.16)

(9.17)

R

The partition function Z is of course exponentially large in N and we are interested in the thermodynamic limit of the free energy F = −kB T lim

N →∞

1 ln Z. N

(9.18)

Therefore we need ln Z = N Sn + ln

∞ j=0

∞ µj λj = N Sn + j (kB T )j j! (k B T ) j! j=1

(9.19)

which implicitly defines the λj in terms of µj . Explicitly we have for the first few terms


¾

λ1 = µ1 λ2 = µ2 − µ21

(9.20) (9.21)

λ3 = µ3 − 3µ1 µ2 + 2µ31 λ4 = µ4 − 4µ3 µ1 − 3µ22 + 12µ2 µ21 − 6µ41 .

(9.22) (9.23)

The µn will contain all powers of N up to N n . However, the λn must contain only terms linear in N because the free energy is extensive. Thus we must have λn = N × coefficient of N in µn .

(9.24)

This is completely analogous to the step in the derivation of the cluster expansion in chapter 6 of the passage from all diagrams to all connected Mayer diagrams. The computation of λn using (9.24) now involves two steps: the integrals over dnR which are easily done, and a combinatorial sum which may be expressed as a graph counting problem. In this chapter we will not explore further details of the relevant graph counting problem and in any event the counting of the graphs must be done by computer. Instead we refer the reader to the original papers where the results are derived and focus instead on the presentation and interpretation of the results of five decades of work. High temperature series expansions for the n vector model are generally written for the specific heat in the form ∞

cH =

q (J/kB T )2 ek (J/kB T )k 2n

(9.25)

k=0

and for the susceptibility ∞

kB T χ(T ) =

1 ak (J/kB T )k n

(9.26)

k=0

where q is the number of nearest neighbors and the normalization is chosen such that e0 = a0 = 1.

(9.27)

The only exception is the Ising model (n = 1) where instead the variable v = tanh J/kB T

(9.28)

is often used as the expansion variable and the expansions are of the form ∞

cH

q = (J/kB T )2 ek v k 2

(9.29)

k=0

where q is the coordination number of the lattice and kB T χ(T ) =

∞

ak v k .

(9.30)

k=0

In practice we are only able to compute a finite number of terms in a high temperature series expansion and thus in order to use these expansions to study critical

¾


phenomena some method extrapolation will be needed. All methods of extrapolation make assumptions which may be open to criticism and thus it is extremely useful that there are two special cases for which exact computations may be carried out: the case n = 1 (the Ising model) in D = 2 and the limit n → ∞ in arbitrary D. The Ising model in D = 2 will be treated in detail in chapters 10–12. The internal energy of the Ising model on the isotropic square lattice is u = −J coth 2J/kB T [1 + 2π −1 (2 tanh2 2J/kB T − 1)K(k)]

(9.31)

with k=2

sinh 2J/kB T (cosh 2J/kB T )2

(9.32)

where K(k), the complete elliptic integral of the first kind is defined as

π/2

K(k) = 0

dφ . (1 − k 2 sin2 φ)1/2

(9.33)

The specific heat is given in terms of the internal energy u as c=

∂u 1 ∂u =− ∂T kB T 2 ∂β

(9.34)

and thus, noting that the complete elliptic integral has the expansion for small k 2 ∞ π (1/2)(1 + 1/2) · · · (n + 1/2) K(k) = k 2n 2 n=0 n!

(9.35)

an exact expression for the high temperature series expansion of the specific heat can be obtained. From the exact expressions (9.31) and (9.34) we find several important properties of the specific heat. We first note that the complete elliptic integral K(k) has a logarithmic singularity at k 2 = 1 and is analytic everywhere else. It thus follows from (9.34) that the specific heat has a logarithmic singularity at k 2 = 1 which corresponds to sinh 2J/kB T = ±1. (9.36) Any method of using a finite number of terms in a high temperature series expansion to extract a critical temperature and critical exponent must be tested against this exact result. We also note that the internal energy depends on J/kB T only through the variable k and thus is a periodic function of J/kB T with period iπ. This feature of periodicity is only a feature of the case n = 1 and does not hold for any other value of n. Consequently when we use the n = 1 model to compare with general values of n we will often expand in terms of J/kB T and not a variable such as v = tanh J/kB T which is only appropriate for n = 1.


¾

The n → ∞ case is called the spherical model and is exactly solvable [1–3]. For D > 2 the model has a positive critical point with exponents α = (D − 4)/(D − 2) γ = 2/(D − 2) for 2 < D < 4

(9.37)

α = 0,

(9.38)

γ=1

D > 4.

The case of D = 2 is particularly interesting. Here if we define the variable k from k J = K(k) kB T 2π

(9.39)

the susceptibility is [4] kB T χ =

π k kB T = . J 4(1 − k) 2(1 − k)K(k)

(9.40)

This susceptibility is analytic for all positive T and diverges exponentially as T → 0. For D = 2 and n = 1 the susceptibility of the square lattice was expanded in 2001 to order 330 by Orrick, Nickel, Guttmann and Perk [5]. For arbitrary n in D = 2 the longest series for the susceptibility on the square lattice is in the 1996 paper of Butera and Comi [6]. On the triangular lattice the longest series for the susceptibility for n = 2, 3 is from the 1996 articles of Campostrini, Pelissetto, Rossi and Vicari [7,8]. For D = 2 and n = 1 the specific heat of the square lattice was exactly computed in 1944 by Onsager [9] and for the triangular lattice in 1950 by Houtapple [10]. For D = 2 and n = 2, 3 the longest series for the specific heat on the square and triangular lattices is from the 1996 articles of Campostrini, Pelissetto, Rossi and Vicari [7, 8]. For D = 3 the longest series for the susceptibility for arbitrary n for cubic and bcc lattices are the 1999 results of Butera and Comi [11]. On the fcc lattice the longest series for the susceptibility in n = 3 and D = 3 are the 1974 results to 10th order of Rushbrooke, Baker and Wood [12] and the 11 and 12 order results done in 1982 of McKenzie, Domb and Hunter [13]. For n = 2 on the fcc lattice the longest series is the 1967 work of Bowers and Joyce [15]. For n = 1 on the bcc lattice the longest series (to order k = 21) was first obtained by Nickel [14] in 1980 and on the fcc lattice the longest series are from the 1975 work of McKenzie [18]. For D = 3 the longest series for specific heats for arbitrary n for cubic and bcc latices are the 1999 results of Butera and Comi [19]. For n = 1 the longest series for the cubic lattice (to order k = 24) was obtained Guttmann and Enting [21,22] in 1994. For the fcc lattice the longest series for the specific heat for n = 1 are the 1972 results of Sykes, Hunter, McKenzie and Heap [20] and for arbitrary n the longest series are the 1979 results of English, Hunter and Domb [23]. These references are summarized in Table 9.1. 9.1.1

Results for D = 2

In chapter 4 we saw that in two dimensions the n vector model with n ≥ 2 has no spontaneous order while in three dimensions there is a positive temperature below which spontaneous order exists. Consequently the physics in two and three dimensions

¾


Table 9.1 Summary of references for high temperature series expansion for the classical n vector model.

D 2 sq

n 1

2 tri 2 sq

1 arb 2 3

2 tri 3 cub bcc 3 cub 3 fcc

2 3 arb

susceptibility Orrick, Nickel Guttmann, Perk 2001 [5]

specific heat Onsager (1944 exact) [9] Houtapple (1950 exact) [10]

Butera, Comi 1996 [6] Campostrini et al 1996 Butera, Comi 1996 [6] Campostrini et al 1996 Campostrini et al 1996 Campostrini et al 1996 Butera,Comi 1999 [11]

[8] [7] [8] [7]

1 arb

Nickel 1980 [14]

1

S. McKenzie 1975 [18]

2 3

Bowers, Joyce 1967 [15] Rushbrooke, Baker, Wood 1973 [12] McKenzie, Domb, Hunter 1982 [13]

Campostrini et al 1996 [8] Campostrini et al 1996 [7] Campostrini et al 1996 [8] Campostrini et al 1996 [7] Butera,Comi 1999 [19] Guttmann, Enting 1993-94 [21, 22] English, Hunter Domb 1979 [23] Sykes,Hunter McKenzie, Heap 1972 [20]

will be quite different and this difference is very apparent in the high temperature expansions. The specific heat for D = 2 has been expanded to order 21 on the square lattice for n = 2, 3, 4, 8 and on the triangular lattice to order 14 by Campostrini, Pelissetto, Rossi and Vicari [7, 8]. The results for n = 2, 3 are given in Table 9.2 and indeed there is a dramatic difference between the case n = 2 where on both the triangular and square lattice all coefficients beginning with e6 are negative and the case n = 3 where on both the square and triangular lattice there is oscillation of signs as k increases. This oscillation of signs is also seen in the results of [8] for n = 4 and 8. The difference is made even more striking when compared with the exact result for the Ising model n = 1 where all coefficients are known to be positive. The susceptibility for D = 2 has been expanded to order 21 on the square lattice for arbitrary n by Butera and Comi [6] and for n = 2, 3, 4, 8 by Campostrini, Pelissetto, Rossi and Vicari [7], [8], and on the triangular lattice to order 15 [7], [8]. The results are given in Table 9.3. In contrast with the specific heat coefficients of Table 9.2 all entries in Table 9.3 are positive. Nevertheless it is expected, that for sufficiently large order, the coefficients for n = 3 will oscillate just as did the coefficients for the specific heat. These oscillations can be inferred by an examination of the coefficients for arbitrary n of Butera and Comi [6] which we reproduce in the appendix and are shown in Figures 9.1 and 9.2 where we plot ak /2k versus k. These results for the n vector model with finite n are to be compared with the exact result (9.40), (9.39) for the susceptibility of the spherical


¾

Table 9.2 Normalized high temperature expansion coefficients ek for the specific heat of n vector model in D = 2. The data for n = 2 are from [8]. The data for n = 3 are from [7].

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

n = 3 Heisenberg square triangular 1 1 0 4 4.2 10.2 0 16 −3.42857 3.77143 0 −84.48 −27.512 −430.664 0 −1607.00 −102.259 −5193.59 0 −14622.4 −206.407 −34794.1 0 −65031.6 140.219 −67930.6 0 137022. 2293.62 1.2417 × 106 0 7755.67 0 17992.4 0 16097.7

n = 2 XY square triangular 1 1 0 4 4.50 10.5 0 20 1.66666 29.1666 0 28 −4.52083 −35.4375 0 −418.777 −54.825 −40.4705 0 −7821. −223.354 −25667.1 0 −77734.3 −684.634 −225152. 0 −633142. −1748.06 −1.7347 × 106 0 −4233.76 0 −8017.15 −210209.4

model which is expanded to order 62 in Table 9.4. In Fig. 9.3 we plot for the spherical model ak (45.484177)k where the normalizing factor is the exact radius of convergence. In Fig. 9.1 we see that for n = 1, 2 there is no indication that ak ever vanishes.We also see that ak is positive for all values of n for k ≤ 10. However, we also see in Fig. 9.2 that for n ≥ 5 negative values of ak occur for k ≤ 21 and it is extremely plausible that negative values of ak also occur for n = 3, 4 at values of k which are not unreasonably large. This behavior of ak as a function of n is very reminiscent of the behavior of the hard sphere virial coefficients as a function of dimension D. From Figs. 9.1–9.3 we infer that for all n > 2 the coefficients ak oscillate in sign as k → ∞ and that as a function of k the location of the first minimum decreases monotonically as a function of n as n increases. For the n → ∞ spherical model limit the period of oscillation in k is 8 which locates the leading singularity of the susceptibility in the complex plane at Reπi/4 where R is the radius of convergence. Unfortunately, the dependence on n (if any) of the period of oscillation cannot be obtained from the data for k ≤ 21.

¾


Table 9.3 Normalized high temperature expansion coefficients ak for the susceptibility of the n = 1, 2, 3 vector model square and triangular lattices in D = 2. The data for the square lattice are from [6–8]. The data for the triangular lattice are from [7, 8]. The entry in the table is to be multiplied by the power of 10 given in parentheses.

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 9.1.2

n = 3 Heisenberg square triangular 1 1 4 6 12 30 33.6 134.4 85.6 560.4 2.06857(2) 2.21974(3) 4.80823(2) 8.45020(3) 1.07756(3) 3.11315(4) 2.34924(3) 1.11528(5) 4.97829(3) 3.89999(5) 1.03249(4) 1.33503(6) 2.09755(4) 4.48333(6) 4.18002(4) 1.47960(7) 8.18813(4) 4.80442(7) 1.57742(5) 1.53661(8) 2.99110(5) 4.84492(8) 5.58647(5) 1.02851(6) 1.86601(6) 3.33902(6) 5.89034(6) 1.02425(7)

n=2 square 1 4 12 34 88 2.19333(2) 529 1.24442(3) 2.86868(3) 6.48988(3) 1.44917(4) 3.19527(4) 6.97111(4) 1.50671(5) 3.23002(5) 6.87193(5) 1.45252(6) 3.05148(6) 6.37548(6) 1.32535(7) 2.74238(7) 5.65014(7)

XY triangular 1 6 30 135 570 2306 9.04147(3) 3.45821(4) 1.29634(5) 4.77988(5) 1.73825(6) 6.24694(6) 2.22202(7) 7.83250(7) 2.73888(8) 9.50902(8)

n=1 Ising square 1 4 12 34.6666 92 2.40533(2) 6.11200(2) 1.53818(3) 3.80964(3) 9.36466(3) 2.28208(4) 5.53176(4) 1.33267(5) 3.19862(5) 7.64296(5) 1.82100(6) 4.32421(6) 1.02449(7) 2.42092(7) 5.71013(7) 1.34401(8) 3.15860(8)

A qualitative interpretation of the D = 2 data

The three different types of behavior of the high temperature series for n = 1, 2, 3 vividly illustrates both the utility and the limitations of high temperature series in the study of phase transitions and critical phenomena. Consider first the Ising model n = 1. Here the coefficients of both the specific heat and the susceptibility do not oscillate in sign as k → ∞ which means that the leading singularity occurs at a positive value of T . This value may be estimated from the series expansions and is found to be the same for both the specific heat and the susceptibility. Moreover because the coefficients, in addition to having no oscillations in sign are all positive the specific heat and the susceptibility both monotonically increase as T decreases and, indeed will diverge at Tc . The nature of these divergences will be studied in detail in chapters 10 and 12. In contrast when n ≥ 3 the coefficients of both the specific heat and the susceptibility oscillate in sign as k → ∞ and therefore the leading singularity occurs not on the positive T axis but in the complex plane. Indeed we see in the spherical model


¾

Table 9.4 High temperature expansion coefficients of the susceptibility ak in D = 2 for the n → ∞ (spherical model) in terms of the variable Jn/kB T . The data are from [4].

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

ak 1 4 12 32 76 160 304 512 748 928 880 512 80 256 2752 8192 12332 5536 −37008 −122368 −178096

k 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

ak −47360 677568 2097152 2918416 407296 −12607296 −37715968 −50921792 −2226176 240405248 703987712 929619628 −29107808 −4684667536 −13519134208 −17550216752 1607717632 92968930496 265456566272 339970952848 −47971688192

k 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

ak −1873359399232 −5306076848128 −6720867874112 1226303567872 382358039040 107615266537472 135063348070288 −29380330962176 −788936558830912 −2209168717692928 −2751199251809600 68130811657216 16430657778461440 45816112359145472 56679265503639232 −155517312752318464 −344954419919964416 −958513093597462528 −11789689975952248323 349764119023599616 7293149768561830912

that there are an infinite number of singularities in the complex T plane and that on the real T axis the free energy and susceptibility only fail to be analytic at T = 0. It is believed that this is the case for all n ≥ 3. It is not possible, at least without strong additional assumptions, to study the behavior of such systems as T → 0 from the high temperature expansions. The feature of having the critical temperature at T = 0 is referred to as being “asymptotically free” and is a property which the n = 3 model in D = 2 shares with gauge theories of strong interactions in four dimensions. We finally consider the case of n = 2. Once again there is no oscillation in the coefficients of either the specific heat or the susceptibility and thus the leading singularity is on the positive real axis. But now the coefficients in the specific heat are negative as k → ∞ so the specific heat, which must be positive, is not monotonic and cannot diverge to infinity at T → ∞. Indeed, further study shows that the singularity is not a power law singularity but instead has a weak essential singularity of exponential form. Furthermore below Tc it follows from the theorems in chapter 4 that unlike the Ising model n = 1 there can be no spontaneous magnetization below Tc and thus the transition at Tc for n = 2 in D = 2 is very different from the typical magnetic transition of the Ising model in D = 2, 3 or the Heisenberg model in D = 3. This new type of transition was first discovered by Kosterlitz and Thouless [24] in 1973.

¾


n=1

50 40

ak /2k

30

n=2

20

n=3

10

n=4 15

10

5

20

k Fig. 9.1 A plot of the normalized coefficients ak /2k of [6] of the susceptibility of the n vector model for n = 1, 2, 3, 4 in D = 2 for 1 ≤ k ≤ 21. The coefficients are reproduced in the appendix.

n=5

n = 10

1.25

n = 20 1

ak /2k

0.75 0.5

n = 200

0.25 12

16

14

18

20

−0.25

k Fig. 9.2 A plot of the normalized coefficients ak /2k of [6] of the susceptibility of the n vector model for n = 5, 10, 20, 200 in D = 2 for 1 ≤ k ≤ 21. The coefficients are reproduced in the appendix.

9.1.3

Results for D = 3

The series expansions for D = 3 are much more uniform than those of D = 2 because the signs of all the coefficients for both the specific heat and the susceptibility are positive for all n. Thus the leading singularity for all cases is on the positive real T axis. Furthermore we saw in chapter 4 that for all n for D = 3 there is spontaneous magnetization at sufficiently low temperatures. Therefore the n vector model for arbitrary n in D = 3 will behave qualitatively as does the n = 1 Ising model in D = 2 and the phenomena of the Kosterlitz–Thouless transition and asymptotic freedom do not exist in D = 3. The results of the series expansions are presented below. The series for the susceptibility of the Ising model is particularly elegant in terms of the variable v because all the expansion coefficients are integers. The results of this expansion are given in


¾

0.08 0.06 0.04 ak (.45484177)k

0.02 10

20

30

40

50

60

−0.02 k

−0.04

Fig. 9.3 A plot of the normalized coefficients ak (.45484177)k of the susceptibility of the spherical (n → ∞) model for 11 ≤ k ≤ 62.

Table 9.5. For comparison with n ≥ 2 where the variable v is not useful we give in Table 9.6 the expansion coefficients in terms of the expansion variable J/kB T . The expansion of the specific heat in terms of v and J/kB T are both used in the literature but unlike the susceptibility expansion the coefficients are not in general integers and because our focus is on comparison with n ≥ 2 the expansion in Table 9.7 is in terms of the variable J/kB T . For n = 2, 3 the numerical values for the susceptibility coefficients are given in Table 9.8. For the cubic and bcc lattices the data are from [11], the fcc data for n = 2 is from [15] and the fcc data for n = 3 is from [12] and [13]. The numerical values for the specific heat coefficients are given in Table 9.9. For the cubic and bcc lattices the data are from [19] and for the fcc lattice the data is from [23]. We note that the Hamiltonian used in the papers of Butera and Comi [11] and [19] is normalized to unity whereas in (9.1) there is a factor of Jn. Thus the critical values βcBC of [11] and [19] are related to our critical variable J/kB Tc by βcBC = n 9.1.4

J . kB Tc

(9.41)

Critical exponents

A major reason for the computation of high temperature series expansions is the desire to use them to study the phase transitions and critical exponents of the system. Logically, of course, this can never be done with only a finite number of terms. However, for systems that have not been solved exactly the extrapolation of high temperature expansions is one of the very few ways in which critical phenomena can be studied. All of the methods by very definition of an extrapolation scheme must make assumptions about the behavior of the function being extrapolated which ultimately cannot be verified with only a finite number of terms. To see the need for making assumptions, consider the magnetic susceptibility at H = 0 which for T > Tc is given in terms of the spin correlation functions as kB T χ(T ) = kB T

∂M (T, H) |H=0 = nz0 nzR ∂H R

(9.42)

¾


Table 9.5 Normalized high temperature expansion coefficients ak using v as the expansion variable for the susceptibility of the n = 1 (Ising) model on the simple cubic, fcc and bcc lattices in D = 3. The data for cubic is from [16, 17], the data for bcc is from [14] and the data for fcc is from [18]. k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

cubic 1 6 30 150 726 3510 16710 79494 375174 1769686 8306862 38975286 182265822 852063558 3973784886 18527532310 86228667894 401225368086 1864308847838 8660961643254

bcc 1 8 56 392 2648 17864 118760 789032 5201048 34268104 224679864 1472595144 9619740648 62823141192 409297617672 2665987056200 17333875251192 112680746646856 731466943653464 4747546469665832 30779106675700312 199518218638233896

fcc 1 12 132 1404 14652 151116 1546332 15734460 159425580 1609987708 16215457188 162961837500 1634743178420 16373484437340 163778159931180 1636328839130860

and the internal energy which is given in terms of the nearest neighbor two-spin correlation as U = −Jnqn0 · nRnn = −Jn2 qnz 0 · nz Rnn (9.43) where Rnn is one of the q nearest neighbors of the origin. The susceptibility χ(T ) diverges as T → Tc and in chapter 5 we defined the critical exponent γ as kB T χ(T ) ∼ Aχ (T − Tc )−γ (9.44) as T → Tc . But from the expansion of the susceptibility in terms of spin correlation (9.42) we see that (9.44) is not the only singularity at Tc present in χ(T ) because, from (9.43), the nearest neighbor correlation will have a singularity, coming from the singularity in the specific heat, of (T − Tc )1−α .

(9.45)

Indeed, from the study of the Ising model in chapters 10–12 we not only expect singularities of the form (9.45) in further neighbor correlations but higher order singularities such as (T − Tc )m(1−α) (9.46) as well.


¾

Table 9.6 Normalized high temperature expansion coefficients ak using J/kB T as the expansion variable for the susceptibility of the n = 1 vector (Ising) model for simple cubic, fcc and bcc lattices in D = 3 as obtained from Table 9.5. The entry in the table is to be multiplied by the power of 10 given in parentheses.

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

cubic 1 6 30 148 706 333600(3) 1.57533(4) 7.37537(4) 3.42619(5) 1.59037(6) 7.34697(6) 3.39206(7) 1.56104(8) 7.18079(8) 3.29549(9) 1.51188(10) 6.92398(10) 3.17010(11) 1.44943(12) 6.62559(12) 3.02535(13) 1.38116(14)

bcc 1 8 56 3.89333(2) 2.61057(3) 1.74731(4) 1.15250(5) 7.59546(5) 4.96669(6) 3.24586(7) 2.11101(8) 1.37234(9) 8.89225(9) 5.75990(10) 3.72216(11) 2.40468(12) 1.55078(13) 9.99874(13) 6.43783(14) 4.14433(15) 2.66495(16) 1.71338(17)

fcc 1 12 132 1400 14564 1.49714(5) 1.52685(6) 1.54836(7) 1.56350(8) 1.57354(9) 1.58301(10) 1.58183(11) 1.58123(12) 1.57843(13) 1.57341(14) 1.56501(15)

It is not unreasonable to expect that similar problems will in general also be present in the specific heat. Thus we expect that as more and more terms are added to a high temperature series expansion eventually more and more of this complicated singularity structure will be needed to explain the data and that any conclusions drawn from a finite number of terms in a high temperature series expansion run the risk of being misleading. Nevertheless many schemes have been used in the past 50 years to estimate critical exponents from high temperature series expansions and the subject is exhaustively and extensively discussed by Guttmann [25–27] with many examples. The most general of these methods is the method of differential approximates which approximates the susceptibility and specific heat by a function F (x) which satisfies a linear inhomogeneous finite order differential equation K k=0

Qk (x)

dk F (x) = P (x) dxk

(9.47)

where x is some suitable variable such as J/kB T or v = tanh J/kB T and Qk (x) and

¾


Table 9.7 Normalized high temperature expansion coefficients ek in terms of the variable J/kB T for the specific heat of the n = 1 (Ising) model on the simple cubic, fcc and bcc lattices in D = 3. The data for cubic and bcc are from [11, 19]. The fcc data are from [20]. The entry in the table is to be multiplied by the power of 10 given in parentheses. k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

cubic 1 0 11 0 1.80600(2) 0 2.74549(3) 0 4.59474(4) 0 7.99761(5) 0 1.42349(7) 0 2.57857(8) 0 4.73436(9) 0 8.78548(10) 0 1.64447(12) 0

bcc 1 0 35 0 9.90667(2) 0 3.10012(4) 0 1.03199(6) 0 3.55827(7) 0 1.25638(9) 0 4.51468(10) 0 1.64433(12) 0 6.05317(13) 0 2.24770(15) 0

fcc 1 8 65 5.33333(2) 4.43067(3) 3.77291(4) 3.28515(5) 2.90479(6) 2.59697(7) 2.34183(8) 2.12669(9) 1.94281(10) 1.78387(11)

P (x) are polynomials. Several special cases of this most general form are extensively used and must be noted: 1) Pad´ e approximants Here K = 0 and the approximating function is F (x) = P (x)/Q0 (x)

(9.48)

with P (x) and Q(x) polynomials. This will be useful if the function being approximated has poles as its only singularities. 2) dlog-Pad´ e Approximants Here K = 1 and Q0 (x) = 0 so the approximating function satisfies P (x) dF (x) = . dx Q1 (x)

(9.49)

This form is useful if the function being approximated is of the form B(x)(1 − x/x0 )−p with B(x) regular at x0 .

(9.50)


¾

Table 9.8 Normalized high temperature expansion coefficients ak for the susceptibility of the n = 2, 3 vector model simple cubic, fccand bcc lattices in D = 3. The entry in the table is to be multiplied by the power of 10 given in parentheses. The data for cubic and bcc are from [11], for fcc with n = 2 from [15], and for fcc with n = 3 from [12] and [13]. Note that the coefficients here for n = 3 and for n = 2 on the cubic and bcc lattices are nk times the coefficients given in the references because of the factor of n in our Hamiltonian (9.1).

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

n = 3 Heisenberg cubic bcc fcc 1 1 1 6 8 12 30 56 132 146.40 387.20 1396.8 690.2 2580.8 14455.2 3.22388(3) 1.70856(4) 1.47439(5) 1.48249(4) 1.11400(5) 1.48869(6) 6.77947(4) 7.23335(5) 1.49196(7) 3.07496(5) 4.65826(6) 1.48668(8) 1.38975(6) 2.99144(7) 1.47468(9) 6.24927(6) 1.91123(8) 1.45734(10) 2.80303(7) 1.21858(9) 1.43578(11) 1.25292(8) 7.74262(9) 1.41087(12) 5.59000(8) 4.91184(10) 2.48777(9) 3.10821(11) 1.10557(10) 1.96376(12) 4.90388(10) 1.23925(13) 2.17245(11) 7.80998(13) 9.61217(11) 4.91466(14) 4.24853(12) 3.09023(15) 1.87561(13) 1.94068(16) 8.27185(13) 1.21803(17)

n=2 cubic 1 6 30 147 696 3.27498(3) 1.51715(4) 7.00091(4) 3.20512(5) 1.46371(6) 6.65206(6) 3.01779(7) 1.36457(8) 6.16220(8) 2.776167(9) 1.24946(10) 5.61341(10) 2.51990(11) 1.12964(12) 5.06076(12) 2.26469(13) 1.01291(14)

XY bcc 1 8 56 388 2592 1.72307(4) 1.12839(5) 7.36260(5) 4.77381(6) 3.08660(7) 1.98584(8) 1.27584(9) 8.16951(9) 5.22557(10) 3.33444(11) 2.12595(12) 1.35332(13) 8.60485(13) 5.46486(14) 3.46878(15) 2.19929(16) 13.9377(17)

fcc 1 12 132 1398 14496 1.48294(5) 1.50307(6) 1.51324(7) 1.51568(8)

3) First order approximants Here K = 1, and Q0 (x) does not vanish identically. These functions are useful if the function being approximated is of the form A(x) + B(x)(1 − x/x0 )−p

(9.51)

with A(x) and B(x) regular at x = x0 . 4) Second order approximants Here K = 2. These functions are useful if the function being approximated is of the form A(x) + B1 (x)(1 − x/x0 )−p1 + B2 (x)(1 − x/x0 )−p2 (9.52) with A(x) and Bk (x) regular at x = x0 . Approximates with K > 2 will allow there to be K − 1 singularities. To use these approximants the orders of the polynomials are chosen such that the coefficients in the polynomials are determined by having the approximating function agree with the finite number of terms computed in the high temperature expansion.

¾


Table 9.9 Normalized high temperature expansion coefficients ek for the specific heat of the n = 2, 3 vector model simple cubic, fcc and bcc lattices in D = 3. The data for cubic and bcc lattices are from [19] and the data for the fcc lattice are from [23]. The entries in the table are to be multiplied by the power of 10 given in parentheses. k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

n = 3 Heisenberg cubic bcc fcc 1 1 1 0 0 8 10.2 34.2 62.4 0 0 511.998 1.50571(2) 8.96573(2) 4.09426(3) 0 0 3.32025(4) 1.94176(3) 2.51956(4) 2.75047(5) 0 0 2.31400(6) 2.81999(4) 7.52999(5) 1.97004(7) 0 0 1.69327(8) 4.30072(5) 2.34097(7) 1.46705(9) 0 0 1.27987(10) 6.74276(6) 7.48567(8) 0 0 1.08142(8) 2.44625(10) 0 0 1.76635(9) 8.13204(11) 0 0 2.92754(10) 2.74057(13) 0 0 4.90992(11) 9.33970(14)

n=2 cubic 1 0 10.4 0 1.61667(2) 0 2.23256(3) 0 3.44788(4) 0 5.57100(5) 0 9.19269(6) 0 1.56205(8) 0 2.68515(9) 0 4.67471(10) 0 8.22264(11)

XY bcc 1 0 34.5 0 9.31667(2) 0 2.73649(4) 0 8.56307(5) 0 2.8417(7) 0 9.29620(8) 0 3.1677(10) 0 1.09565(12) 0 3.83781(13) 0 1.35783(15)

fcc 1 8 64.5 520 4.21667(3) 3.49162(4) 2.95292(5) 2.53723(6) 2.20588(7) 1.93569(8) 1.71171(9)

The critical point is then determined by the smallest value of 1/T at which the approximating function is singular. This singularity is at the zeros of the leading coefficient QK (x). At this singularity the approximating function in general has power law singularities which are easily determined from the indicial equation of the linear differential equation (9.47). We thus see that in principle the susceptibility needs to be approximated by at least a second order approximate in order to encompass the singularities γ and 1 − α. High temperature series are one of the very few tools available to study critical phenomena for systems which are not exactly solvable and thus they are widely used to gain insight into critical phenomena. Nevertheless, the validity of the exponents obtained is in the last analysis a subjective decision to be made by the reader. The exponents γ and α will be estimated below in section 1.6 using the various cases of differential approximates. However, to gain a qualitative insight into the analysis we begin with a discussion of the most primitive of all methods; the ratio method.

9.1.5

The ratio method

The most elementary way to study the radius of convergence and leading singularity in the high temperature series expansion is by analysis of the ratios of coefficients. The method begins with the elementary expansion valid for p > 0

Classical n vector model for D = 2, 3 ∞ k=0

(x/x0 )k

Γ(k + p) = [1 − (x/x0 )]−p . k!Γ(p)

¾

(9.53)

Thus calling ck the coefficient of xk we see that as k → ∞ p−1 ck → x−k /Γ(p)] 0 k

(9.54)

and thus (p−1) −2 rk = ck+1 /ck = x−1 ∼ x−1 )]. 0 (1 + 1/k) 0 [1 + (p − 1)/k + O(k

(9.55)

The simplest version of the ratio test uses this simple relation to estimate the critical value x0 as 2 krk − (k − 1)rk−1 = x−1 (9.56) 0 [1 + O(1/k )] or, more generally in a form which reveals some of the arbitrariness inherent in the method 2 (k + )rk − (k + − 1)rk−1 = x−1 (9.57) 0 [1 + O(1/k )]. The exponent p is then estimated (with = 0) as p=

k(2 − k)rk − (k − 1)2 rk−1 [1 + O(1/k)]. krk − (k − 1)rk+1

(9.58)

The estimates (9.56)–(9.58) are known as unbiased estimates. If the critical value x0 is known the exponent p may be determined from pk = k{x0 (ck+1 /ck ) − 1} + 1 + O(1/k).

(9.59)

This estimation method is referred to as a biased estimate. Ratios for D = 3 The qualitative behavior of the critical phenomena for D = 3 is the same for all values of n in the sense that for all n the critical temperature is greater than zero. Consequently, even though in the presentation of the results for the Ising model (n = 1) it was most natural to use the variable v = tanh J/kB T for which the expansion coefficients of the susceptibility (as seen in Table 9.5) were positive integers we will here use the variable J/kB T as the expansion variable for n = 1 as well as for n ≥ 2 In Table 9.10 we give the ratio ak+1 /ak q for the coefficients of the susceptibility expansion in terms of the variable J/kB T normalized by the number of nearest neighbors q for the n = 1, 2, 3. For convenience we restrict our analysis to the bipartite cubic and bcc lattices for which the series are substantially longer than the closest packed fcc lattice. The ratios scaled by q are plotted in Fig. 9.4 for n = 1, 2, 3 for the cubic lattice. We see here that there is a small odd-even oscillation which is due to the fact that for the bipartite lattices that in addition to the singularity at T = Tc , there is an additional (antiferromagnetic) singularity at T = −Tc due to the symmetry (9.12). This oscillation rapidly decreases as k increases and the points rapidly approach a straight line.

¾


Table 9.10 The ratios rk = ak+1 /ak q for the susceptibility kB T χ(T ) in terms of the expansion variable J/kB T of the n = 1, 2, 3 vector model for D = 3 for the cubic and bcc lattices. e,o are obtained by a linear extrapolation (9.60). The values for r∞

k 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 e r∞ o r∞

ak+1 /ak q cubic .822222 .795045 .787535 .787035 .780299 .774241 .773634 .769943 .769492 .767007 .766667 .764886 .764621 .763286 .763073 .762032 .761862 .761027 .760882 .75207 .75248

for n = 1 bcc .869047 .838155 .836652 .824482 .823803 .817378 .816907 .812963 .812609 .809953 .809680 .807776 .807554 .806126 .805944 .804830 .804683 .803794 .803664 .79450 .79599

ak+1 /ak q cubic .816667 .789116 .784239 .772091 .769087 .763024 .761131 .757443 .756104 .753625 .752642 .750859 .749991 .748898 .748179 .747146 .746663 .745833 .745437 .73440 .73467

for n = 2 bcc .866071 .835052 .830956 .818590 .815609 .810483 .808212 .804218 .803086 .800405 .799554 .797626 .796967 .795715 .794791 .793863 .793428 .792530 .792170 .78084 .78119

ak+1 /ak q cubic .813333 .785747 .778489 .766411 .762172 .755949 .753262 .749447 .747536 .744980 .743596 .741732 .740804 .739135 .738344 .737429 .736658 .735789 .735036 .72043 .72184

for n = 3 bcc .864286 .833161 .827534 .815014 .811642 .804997 .802725 .798625 .796948 .794226 .792987 .790999 .789747 .788825 .787773 .786599 .785972 .785006 .784538 .77163 .77146

The limiting value for k → ∞ is estimated by linearly extrapolating the even and odd values of k using 1 r∞ = {(k + 2)rk+2 − krk } (9.60) 2 and the exponent γ is estimated from (9.59) using the estimator γk = 1 + k(

rk − 1) + O(1/k) with rk = ak+1 /ak q. r∞

(9.61)

e,o we give the average of the extrapolations In Table 9.10 in the entries marked r∞ obtained from (9.60) using the values at k = 20, 18 and k = 19, 17. In Table 9.11 we give the estimators (9.61) for the exponent γ. There we see that there is serious discrepancy between the odd and even estimators and that the cubic and bcc exponents are not in accord with the expectations of universality. The conclusion is that, particularly with the odd-even oscillation, the ratio method is far too crude to be used for quantitative results for critical exponents. The ratio method is applied to the specific heat for the cubic and bcc lattices by computing in Table 9.12, for even k, the ratios scaled by rkα = ek+2 /ek q 2 from the


¾

0.78 0.77 n=1

0.76 ak+1 /6ak

0.75

n=2

0.74

n=3

0.73

0.02 0.04 0.06 0.08

0.1

0.12 0.14

1/k Fig. 9.4 A plot for the cubic lattice with n = 1, 2, 3 of the ratios of the susceptibility coefficients normalized by q the number of nearest neighbors ak+1 /6ak versus 1/k. k Table 9.11 The estimators γk = 1 + k( rr∞ − 1) for the susceptibility exponent γ of the n = 1, 2, 3 vector model for D = 3 for the cubic and bcc lattices.

k 20 19 18 17

γk for cubic 1.2343 1.2271 1.2343 1.2157

n=1 bcc 1.2305 1.1862 1.2307 1.1887

γk for cubic 1.3003 1.2886 1.3004 1.2886

n=2 bcc 1.2860 1.2758 1.2862 1.2757

γk for cubic 1.4054 1.3671 1.4054 1.3671

n=3 bcc 1.3345 1.3220 1.3345 1.3336

√ α coefficients of Table 9.9. These ratios are ploted in Fig. 9.5. When the values of r∞ in e,o the last line are compared to the values of r∞ of Table 9.10 we see that the agreement is not better than 1 or 2 percent even though they should be the same. This is another indication of the crudity of the ratio method. The difficulties and the arbitrariness in the use of the ratio method are well illustrated by an attempt to estimate the exponent α from the ratios of Table 9.12 by use of an estimator analagous to (9.61) used for the susceptibility k rkα (1) αk = 1 + − 1 (9.62) α 2 r∞ or (2) αk

k =1− 2

α r∞ −1 rkα

(9.63)

We first note that these two estimators, which must give identical results in the limit k → ∞, are substantially different for the data in Table 9.12. Furthermore the esti-

¾


Table 9.12 The ratio rkα = ek+2 /ek q 2 as obtained from Tables 9.7 and 9.9 for the specific heat of the n = 1, 2, 3 vector model for D = 3 for the cubic and bcc lattices. The values √ α α for r∞ are obtained by a linear extrapolation from k = 16, 18. The values of r∞ are to be e,o compared with the corresponding values of r∞ in Table 9.10.

k 2 4 6 8 10 12 14 16 18 α r∞ √ α r∞

n=1 cubic bcc .45606 .44226 .42227 .48895 .46487 .52013 .48350 .53874 .49441 .55169 .50317 .56146 .51001 .56909 .51546 .57560 .51994 .58019 .5557 .6169 .7454 .7854

n=2 cubic bcc .43180 .41006 .38360 .45893 .43464 .48894 .44882 .51852 .45836 .51114 .47200 .53242 .47749 .54044 .48256 .54730 .48860 .55281 .5369 .5968 .7328 .7725

n=3 cubic bcc .41005 .40961 .35822 .43909 .40341 .46697 .42383 .48576 .43550 .50640 .44550 .51060 .45371 .51942 .46038 .52657 .46587 .53249 .5097 .5798 .7139 .7614

α mators are quite sensitive to the value of r∞ , and there are significant differences, if instead of the value obtained from the specific heat ratios in Table 9.12, we used the value obtained from the susceptibility ratios of Table 9.10. If we compute the estimaα o2 o2 tor α(2) using r∞ = r∞ with r∞ obtained from Table 9.12 we obtain the estimates in Table 9.13. (2)

Table 9.13 The estimators αk (9.63) for k = 16, 18 and the extrapolant 9α18 − 8α16 for α o2 o = r∞ where r∞ is obtained from Table 9.12. the cubic lattice using r∞

(2)

9α18 −

(2) α18 (2) α16 (2) 8α16

n=1 0.1988 0.2121 0.0923

n=2 0.0580 0.0586 0.0529

n=3 −0.0660 −0.0543 −0.1600


¾

0.55 0.525 n = 1 0.5 n = 2

ek+2 /36ek 0.475

n=3

0.45 0.425 0.4 0.375 0.05

0.1

0.15

0.2

0.25

1/k Fig. 9.5 A plot for the cubic lattice with n = 1, 2, 3 of the ratios of the specific heat coefficients normalized by q 2 the number of nearest neighbors ek+2 /ek 36 versus 1/k.

Ratios for D = 2 The ratios for the susceptibility for D = 2 are given in Table 9.14 and plotted in Fig 9.6 for n = 1, 2, 3. We see in the figure that for n = 3 there is no linearity to be observed in the plot. This is to expected from the observation that the coefficients are expected to oscillate in sign for sufficiently large k and thus the true large k behavior of the ratios is not expected to be seen in the behavior of the first 20 ratios. 1.3

1.2

n=1 ak+1 /ak

1.1

n=2 1

0.9

n=3 0.05

0.1

0.15

0.2

1/k Fig. 9.6 A plot for the square lattice with n = 1, 2, 3 of the ratios of the susceptibility coefficients ak+1 /ak versus 1/k.

¾


Table 9.14 The ratios ak+1 /ak for the susceptibilities kB T χ(T ) of the n = 1, 2, 3 vector model for the D = 2 square and triangular lattices computed from the data of Table 9.3.

k 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

9.1.6

n=1 square 1.44444 1.32692 1.30725 1.27051 1.25833 1.23836 1.22907 1.21845 1.21200 1.20456 1.20008 1.19473 1.19129 1.18732 1.18460 1.18152 1.17933 1.17686 1.17507

square 1.4166 1.2941 1.2462 1.2059 1.1762 1.1525 1.1312 1.1164 1.1024 1.0908 1.0806 1.0718 1.0637 1.0568 1.0504 1.0446 1.0394 1.0340 1.0301

n=2 triangular 2.2500 2.1111 2.0226 1.9604 1.9124 1.8742 1.8436 1.8183

square 0.9333 0.8492 0.8055 0.7748 0.7470 0.7267 0.7063 0.6913 0.6771 0.6642 0.6529 0.6421 0.6323 0.6225 0.6139 0.6047 0.5964 0.5880 0.5795

n=3 triangular 1.4333 1.3898 1.3203 1.2689 1.2280 1.1941 1.1656 1.1410

Estimates from differential approximates

The best estimates of the exponents γ and α come from the analysis of the high temperature series expansions by means of the differential approximants discussed in section 9.1.4 and many such studies have been made. Early studies used Padé approximants (9.48) and dlog-Padé approximants (9.49). However, as the length of the series was increased and the necessity of accounting for confluent singularities became apparent, it became clear that these extrapolation techniques are not sufficient. We present in Table 9.15 the results of the analysis of [11, 19] of the series for the cubic and bcc lattices which uses first and second order approximants and is thus capable of including one confluent singularity. Even with this tool there are many variations. In particular the orders of the polynomials Qk (x) and P (x) can be freely chosen and each choice will give a different exponent at the critical point. No one choice is preferred and in practice many different choices are made and the spread of the resulting exponents is taken as a measure of the accuracy of the estimate. Furthermore one may compute Tc from the specific heat and susceptibility series separately or the assumption can be made that the Tc for both must be the same and use the value which is considered the most accurate. Such extrapolations which use extra assumptions beyond the series themselves are said to be biased. In [11,19] results

Quantum Heisenberg model

¾

Table 9.15 The estimates of [11, 19] for the critical temperatures, susceptibility exponent γ and the specific heat exponent α for the n vector model with n = 1, 2, 3. The error in the last digit of the estimate is given in the parenthesis.

βc βc γ γ α α

n cub bcc cub bcc cub bcc

1 0.221663(9) 0.157379(2) 1.244(3) 1.243(2) 0.103(8) 0.105(9)

2 0.227095(2) 0.160214(2) 1.327(4) 1.322(3) −0.014(9) −0.019(8)

3 0.23101(1) 0.162268(1) 1.404(4) 1.396(3) −0.11(2) −0.13(2)

of several such additional assumptions are presented along with unbiased estimates and all results are in good agreement. The estimates presented in Table 9.15 are the unbiased estimates reported in [11, 19]. The number in parentheses represents a subjective estimate of the uncertainty in the result for the exponent. We have defined βc = J/kB Tc and note that because the Hamiltonian (9.1) has a different normalization from [11, 19] that 1 = βc = (βc )BC /n qr∞

(9.64)

where (βc )BC is the value of βc in Table II of [11] and ak+1 k→∞ ak

qr∞ = lim

(9.65)

We will discuss the significance of these estimates of the critical exponents in section 9.3 after the corresponding results for the quantum case have been derived.

9.2


High temperature expansions for the Heisenberg model may be developed using the same techniques as were used for the classical n vector model with the replacement dnR → Tr (9.66) R

where the trace is over all states of the system and we note that at infinite temperature the partition function reduces to Z = Tr1 = (2S + 1)N .

(9.67)

The details of the resulting graphical expansion are given in the article of Baker, Gilbert, Eve and Rushbrooke [28]. We will here present the results of the computations using the notation kB T χ(T ) =

∞ k=0

ak (J/kB T )n

(9.68)

¾


kB T χs (T ) =

∞

ask (−J/kB T )n

(9.69)

k=1 ∞

cH /kB =

3q (J/kB T )2 ek (J/kB T )n 2

(9.70)

k=0

where q is the coordination number (number of nearest neighbors) of the lattice. For D = 2 on both the square and triangular lattices the most extensive results known for arbitrary S for both the susceptibility and specific heat are given in the 1958 work of Rushbrooke and Wood [29] as extended in 1968 by Stephenson, Pirne, Wood and Eve [30]. For S = 1/2 the most extensive data for the specific heat and susceptibility are in the 1996 work of Oitmaa and Bonilla [33]. For S = 1/2 the most extensive data for the staggered susceptibility on square lattice are in the 2000 work of Pan [31]. For D = 3 on the cubic, bcc and fcc lattice the most extensive results known for arbitrary S for both the susceptibility and specific heat are given in the 1958 work of Rushbrooke and Wood [29] as extended in 1968 by Stephenson, Pirne, Wood and Eve [30], and the most extensive data for the staggered susceptibility on the cubic and bcc lattice in in the 1963 work of Rushbrooke and Wood [32]. For S = 1/2, 1, 3/2 on the cubic and bcc lattices the most extensive data for the susceptibility and staggered susceptibility is the 2004 work of Oitmaa and Zheng [34]. These references are summarized in Table 9.16 Table 9.16 Summary of references for high temperature series expansion for the quantum spin S Heisenberg model.

D 2 sq tri

S arb.

2 sq tri 3 cub bcc fcc

1/2 arb

χ Rushbrook, Wood 1958 [29] Stephenson, Pirne Wood, Eve 1968 [30] Oitmaa, Bonilla 1996 [33] Rushbrooke, Wood 1958 [29]

χs

Pan 2000 [31]

Stephenson, Pirne Wood, Eve 1968 [30] 3 cub bcc 3 cub bcc fcc 3 cub bcc

arb

specific heat Rushbrooke, Wood 1958 [29] Stephenson, Pirne Wood, Eve 1968 [30] Oitmaa, Bonilla 1996 [33] Rushbrooke, Wood 1958 [29] Stephenson, Pirne Wood, Eve 1968 [30]

Rushbrooke, Wood 1963 [32]

1/2

Oitmaa, Bonilla 1996 [33]

1/2,1,3/2

Oitmaa, Zheng 2004 [34]

Oitmaa Bonilla 1996 [33] Oitmaa, Zheng 2004 [34]


9.2.1

¾

Results for D = 2

For S = 1/2 the most extensive results known are the 14 term expansions of Oitmaa and Bornilla [33] for the square (sq) and triangular (tri) lattices in two dimensions. The results are given in Table 9.17 where we use the notation

en =

eñ (1/2) 4n n!

and an (1/2) =

a ñ (1/2) . 4n n!

(9.71)

It seems that series for the staggered susceptibility and for S ≥ 1 are not known. Table 9.17 High temperature expansion coefficients eñ (1/2) and a ñ (1/2) for the specific heat and susceptibility of the spin 1/2 quantum Heisenberg model for the square and triangular lattices in D = 2 from Table 1 of [33]. We have set en (1/2) = eñ (1/2)/(4n n!) and an (1/2) = a ñ (1/2)/(4n n!). Note that the coupling J of [33] is one half of the J˜ of (9.4) and ek is ek+2 of [33]. n 0 1 2 3 4 5 6 7 8 9 10 11 12 n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Specific heat eñ (1/2) square 1 −2 −14 20 520 −41, 552 153,488 1,437,9136 −205, 339, 264 −6, 828, 335, 360 231,955,086,080 349,717,905,356 −299, 728, 629, 046, 272 Susceptibility a ñ (1/2) square 1 4 16 64 416 4, 544 23,488 −207, 616 4, 205, 056 198, 295, 552 −2, 574, 439, 424 −112, 886, 362, 112 3, 567, 419, 838, 464 94,446,596,145,152 −1, 798, 371, 774, 277, 632

eñ (1/2) triangular 1 2 −34 −360 5464 162, 960 −1, 514, 000 −130, 296, 448 8, 361, 856 154, 693, 752, 576 2,047,410,296,064

a ñ (1/2) triangular 1 6 48 408 3,600 42,336 781,728 13,646,016 90,893,568 −1, 798, 204, 416 70,794,720,768 7,538,546,211,840 6,813,109,782,528

¾

9.2.2


Results for D = 3

For the specific heat with S = 1/2 the most extensive results known are the 14-term expansions of Oitmaa and Bornilla [33] for the simple cubic (sc), bcc and fcc lattices. The results for en are given in Table 9.18 where we use the notation of (9.71). There seem to be no results known for S ≥ 1. Table 9.18 Normalized High temperature expansion coefficients eñ (1/2) for the specific heat of the spin 1/2 quantum Heisenberg model for simple cubic, face centered cubic and body centered cubic lattices in D = 3 from table 1 of [33] We have set en (1/2) = eñ (1/2)/(4n n!).Note that the coupling J of [33] is one half of the J˜ of (9.4) and ek is ek+2 of [33]. n 0 1 2 3 4 5 6 7 8 9 10 11 12 n 0 1 2 3 4 5 6 7 8 9 10

eñ (1/2) for simple cubic 1 −2 −18 280 3,688 −113, 232 −867, 216 80,440,192 288, 502, 656 −95, 126, 989, 944 709,294,331,648 150,744,103,377,920 −3, 074, 209, 362, 326, 528 eñ (1/2) for fcc 1 6 10 −280 9,000 809,200 31, 291, 856 974, 702, 208 4,168,957,2736 3,147,043,161,856 276,332,034,732,800

eñ (1/2) for bcc 1 −2 14 95 2,040 −24, 752 2,334,768 −44, 473, 472 3, 429, 683, 056 −41, 940, 628, 224 4,416,784,659,712 −153, 284, 724, 083, 712 20, 508, 575, 418, 559, 488

For the susceptibilities χ(T ) and χs (T ) with S = 1/2, 1, 3/2 the best results for cubic, bcc, and fcc are those of Oitmaa and Zheng [34]. We present these in Tables 9.19– 9.22 where we use the notation an (1/2) =

asn (1/2) =

a ñ (1/2) 2˜ an (1) 5˜ an (3/2) , an (1) = n+1 , an (3/2) = n+2 n+1 4 n! 3 n! 2 n!

a ˜sn (1/2) , n+1 4 (n + 1)!

asn (1) =

2˜ asn (1) , n+1 3 (n + 1)!

asn (3/2) =

5˜ asn (3/2) . n+3 2 (n + 1)!

(9.72)

(9.73)


¾

Table 9.19 High temperature expansion coefficients for the susceptibilities χ(T ) and χs (T ) of the spin 1/2 quantum Heisenberg model for cubic and bcc lattice from Table 1 of [34]. We have set an (1/2) = a ñ (1/2)/(4n+1 n!) and asn (1/2) = a ˜sn (1/2)/(4n+1 (n + 1)! n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

9.2.3

a ñ (1/2) for the cubic lattice 1 6 48 528 7,920 149,856 3,169,248 77,046,528 2,231,209,728 71,938,507,776 2, 446, 325, 534, 208 92, 886, 269, 386, 752 3,995,799,894,239,232 180,512,165,153,832,960 8, 443, 006, 907, 441, 565, 696 a ñ (1/2) for the bcc lattice 1 8 96 1664 36,800 1,008,768 32,626,560 1,221,399,040 51,734,584,320 2, 459, 086, 364, 672 129,082,499,311,616 7,432,690,738,003,068 464,885,622,793,134,080 31,456,185,663,820,136,448 2,284,815,238,218,471,260,160

a ˜sn (1/2) for the cubic lattice 1 12 168 2,880 59,376 1,478,592 42,537,024 1,353,271,296 48,089,027,328 1,908,863,705,088 83,357,870,602,752 3,926,123,179,720,704 198,436,560,561,973,248 10,823,888,709,015,846,912 635,114,442,481,347,244,032 a ˜sn (1/2) for the bcc lattice 1 16 320 8,192 248,768 8,919,296 367,854,720 17,216,475,136 899,434,884,096 51,925,815,320,576 3,280,345,760,086,016 225,270,705,859,919,872 16,704,037,174,526,894,080 1,330,557,135,528,577,925,120 113,282,648,639,921,512,955,904

Analysis of results

The first observation to be made about the high temperature series expansion of the quantum Heisenberg model for both the specific heat and the susceptibility is that the signs of the coefficients appear to be significantly more irregular than in the classical case. In the case of D = 2, Table 9.17, the series for the specific heat shows the same sort of oscillations in sign as seen in the classical case and the susceptibility for the square and triangular lattices have minus signs at orders 7 and 9 respectively whereas in the classical case minus signs were not seen in the susceptibility up to order 21 and were inferred at higher order only by an analysis of n vector models with n ≥ 3. It is believed that the quantum Heisenberg model in D = 2 shares with the classical model in D = 2 the property of having a phase transition only at T = 0 which is consistent with the leading singularities of the specific heat and susceptibility having a complex value. It is thus not to be expected that information about the phase transition can be

¾


Table 9.20 High temperature expansion coefficients for the susceptibility χ(T ) of the spin 1/2 quantum Heisenberg model for the fcc lattice from Table 2 of [34]. We have set an (1/2) = a ñ (1/2)/(4n+1 n!) n 1 2 3 4 5 6 7 8 9 10 11 12

a ñ (1/2) for fcc lattice 12 240 6,624 234,720 10,208,832 526,810,176 31,434,585,600 2,127,785,02,024 161, 064, 469, 168, 128 13,483,480,670,745,600 1,237,073,710,591,635,456 123,437,675,536,945,410,048

obtained from the series expansion without further assumptions. In this respect the classical and quantum spin 1/2 Heisenberg models are similar in D = 2. For D = 3, however, there is much less similarity between the classical and the quantum case. In the classical case the coefficients in the specific heat expansion are all positive while from Table 9.18 we see that there are many minus signs for the cubic and bcc lattices and there is even one isolated minus sign in the fcc lattice. The oscillations of sign for the cubic and bcc lattices make it fruitless to attempt an analysis of the series with the assumption that the leading singularity is on the positive temperature axis. On the other hand Tables 9.19–9.22 for the direct and staggered susceptibility have only positive signs so an assumption of a leading singularity at a positive temperature is not precluded even though such analysis could not be done for the specific heat. The simplest such analysis is the ratio method. This has been done by Oitmaa and Zheng [34] and we plot these ratios for both the cubic and the bcc lattice for S = 1/2, 1 and ∞ in Fig. 9.7. From this figure we see that aside from the small oscillation between even and odd ratios which was seen even in the classical case that, in all cases except S = 1/2 on the cubic lattice, the ratios seem to be linear in 1/n for large n. The intercept of this plot at 1/n = 0 estimates kB Tc /(JS(S + 1)) and clearly shows that the critical temperature of the staggered (antiferromagnetic) susceptibility is larger than the critical temperature of the direct (ferromagnetic) susceptibility. The slope gives the critical exponent γ appears to be independent of S and is the same for both the direct and staggered susceptibility. A more quantitative estimate of the critical temperature and exponents γ is made in [34] by use of a dlog-Padé analysis. The resulting estimates for the scaled critical temperature kB Tc /(JS(S + 1)) are given in Table 9.23. The estimates of the exponent γ are given in Table 9.24 where we give the range of values obtained by the use of different degrees of the polynomials Q1 (x) and P (x) used in the dlog-Padé analysis.

Discussion

¾

Table 9.21 High temperature expansion coefficients for the susceptibilities χ(T ) and χs (T ) of the spin 1 quantum Heisenberg model for cubic and bcc lattice from Table 3 of [34]. We have set an (1) = 2˜ an (1)/(3n+1 n!) and asn (1) = 2˜ asn /(3n+1 (n + 1)!) n 0 1 2 3 4 5 6 7 8 9 10 11 12 n 0 1 2 3 4 5 6 7 8 9 10 11 12

9.3

a ñ (1) for the cubic lattice 1 12 222 5,904 201,870 8,556,912 426,905,802 2,467,414,724 1,616,505,223,518 118,701,556,096,392 9, 628, 527, 879, 611, 262 856, 813, 238, 084, 411, 136 82,856,991,914,713,902,402 a ñ (1) for the bcc lattice 1 16 424 16,512 819,240 50,363,136 3,652,143,480 307,454,670,000 29,310,549,057,000 3, 133, 368, 921, 937, 824 370,060,173,560,963,304 47,968,071,364,509,850,944 6,756,542,767,252,059,234,840

a ˜sn (1) for the cubic lattice 1 24 702 26,280 1,184,526 63,357,984 3,887,604,666 270,348,199,128 20,988,390,679,758 1,802,403,961,243,776 169,418,364,565,523,958 17,314,303,199,655,636,792 a ˜sn (1) for the bcc lattice 1 32 1320 71136 4,588,968 351,263,232 30,873,601,080 3,082,065,903,648 343,320,789,071,016 42,320,100,429,654,912 5,709,664,512,091,086,984 837,942,419,330,764,322,976

Discussion

For the classical n vector model in three dimensions with n = 1 (Ising), n = 2 (XY) and n = 3 (Heisenberg) the estimates of Table 9.15 obtained from the 31-term series expansions as extrapolated by differential approximants are the most unbiased and systematic computations of the critical exponents α and γ which exist. In particular the agreement within the error bars of the exponents for the cubic and bcc lattices may be taken as good evidence for the hypothesis of universality. Similarly the agreement of the exponents γ for the spin S Heisenberg magnet of Table 9.24 with the classical value of Table 9.15 may be taken as good evidence of the extension of universality from the classical to the quantum system. Using the assumptions of scaling presented in chapter 5 we may compute all other critical exponents in terms of the high temperature exponents α and β estimated in Table 9.15 for the classical model, by use of equations (5.78)–(5.82) of chapter 5. These estimates of the critical exponents are summarized in Table 9.25. Therefore from one point of view it can be argued that we understand a great deal of the physics of the quantum magnet as well.

¾


Fig. 9.7 Ratio plots from [34] for the ferromagnetic and antiferromagnetic susceptibilities for the cubic lattices for S = 1/2, 1, 3/2. The ratios plotted are rn = an /(S(S + 1)an−1 ) where the an are given in Tables (9.19), (9.21) for S = 1/2 and 1 and from the classical Heisenberg model for S = ∞. The ferromagnetic cases are given by the solid lines and the antiferromagnetic cases by the dashed lines. The spin S = 1/2 is given by crosses, S = 1 by diamonds and S = 3/2 by solid circles.

Fig. 9.8 Ratio plots from [34] for the ferromagnetic and antiferromagnetic susceptibilities for the bcc lattices for S = 1/2, 1, 3/2. The ratios plotted are rn = an /(S(S + 1)an−1 ) where the an are given in Tables (9.19), (9.21) for S = 1/2 and 1 and from the classical Heisenberg model for S = ∞. The ferromagnetic cases are given by the solid lines and the antiferromagnetic cases by the dashed lines. The spin S = 1/2 is given by crosses, S = 1 by diamonds and S = 3/2 by solid circles.

Discussion

¾

Table 9.22 High temperature expansion coefficients for the susceptibilities χ(T ) and χs (T ) of the spin 3/2 quantum Heisenberg model for cubic and bcc lattice from [34]. We have set an (3/2) = 5˜ an (3/2)/(2n+2 n!) and asn (3/2) = 5˜ asn (3/2)/(2n+3 (n + 1)!).

n 0 1 2 3 4 5 6 7 8 9 n 0 1 2 3 4 5 6 7 8 9

a ñ (3/2) for the cubic lattice 1 60 1440 50,136 2,241,660 124,125,372 8,102,868,414 613,292,153,184 52,599,376,466,556 50,561,988,998,505,288 a ñ (3/2) for the bcc lattice 1 80 2,720 136,448 8,751,600 696,028,496 65,331,028,472 7,121,212,898,544 879,298,191,968,624 121, 768, 840, 349, 153, 216

a ˜sn (3/2) for the cubic lattice 2 60 2220 106,032 6,103,230 417,121,164 32,715,943,017 2,911,926,450,048 2,289,263,779,556,198 31,792,485,934,519,488 a ˜sn (3/2) for the bcc lattice 2 80 4160 283,776 23,240,440 2,263,139,152 253,095,247,076 32,175,304,799,424 4,563,926,306,507,096 716,734,730,963,510,496

Table 9.23 Estimate of kB Tc /(JS(S + 1)) by use of a dlog-Padé analysis from [34].

S = 1/2 χ χs 1.119(2) 1.259(2) S = 1/2 χ χs 1.6803(6) 1.8350(5)

cubic S=1 χ χs 1.2994(5) 1.3676(7) bcc S=1 χ χs 1.894(1) 1.967(1)

S = 3/2 χ χs 1.37(2) 1.404(7)

S=∞ χ 1.4429

S = 3/2 χ χs 1.97(1) 2.009(6)

S=∞ χ 2.0542

On the other hand this analysis has made many assumptions which are untested and in some cases contradictory. In particular consider the result shown in Table 9.23 that for the quantum spin S Heisenberg model the critical temperature of the ferromagnetic susceptibility lies below the critical temperature of the antiferromagnetic susceptibility. This is troubling because these singularities are not independent of each other and, for example, the oscillation between even and odd coefficients of the direct susceptibility is caused by

¾


Table 9.24 Estimate of γ by use of a dlog-Padé analysis from [34]. We give here the range of estimates of γ obtained by use of different degrees of the approximating polynomials Q1 (x) and P (x).

χ χs S = 1/2 cubic 1.411–1.421 1.440–1.425 S = 1/2 bcc 1.416–1.423 1.431–1.436 S = 1 cubic 1.406–1.411 1.409–1.417 S = 1 bcc 1.398–1.404 1.390–1.405 Table 9.25 Estimates of the critical exponents in D = 3 for the classical n vector model and the quantum Heisenberg model from Table 9.15 and the exponent equalities (5.78)–(5.82) of chapter 5. The values of α and γ for n = ∞ are the spherical model values from (9.37).

exponent α γ β δ ν η ∆

1 0.10 1.24 0.33 4.6 0.63 0.042 1.57

n 2 3 −0.02 −0.12 1.32 1.40 0.35 0.36 4.8 4.9 0.67 0.71 0.039 0.019 1.67 1.76

∞ −1 2 1/2 15 1 0 5/2

the existence of a second singularity at −Tc which, by sending nR → −nR on one of the sublattices of the cubic or bcc lattices, is equivalent to the singularity in the antiferromagnetic susceptibility. For the classical case the existence of this second singularity causes no harm, but for the quantum case where the critical temperature of the ferromagnet is less than the critical temperature of the antiferromagnet this means that ultimately for sufficiently large order the coefficients of the ferromagnetic susceptibility must oscillate between positive and negative values as the order of the coefficient varies from even to odd. There is no trace seen in the data of Tables 9.19, 9.21, and 9.22 of this necessary oscillation in the coefficients of the ferromagnetic susceptibility. Doubt can therefore be cast on the assumption that the series for the quantum Heisenberg magnet are sufficiently long for the true asymptotic behavior to be seen. This is indeed in agreement with the observation already made that the alteration in signs of the specific heat coefficients for the cubic and bcc lattices of Table 9.18 indicates that these series are not long enough to be in the asymptotic regime where the critical exponent α can be obtained. Worse still is the fact that since the physical singularity for the ferromagnet is not

Statistical mechanics versus quantum field theory

¾

leading many more terms in the series expansion for the ferromagnetic case should be needed than for the antiferromagnetic case. Furthermore when we recall that in Fig. 9.7 the coefficients for the S = 1/2 Heisenberg magnet on the cubic lattice were certainly not smooth and from chapter 4 that for the quantum Heisenberg magnet the proof of spontaneous order exists for the antiferromagnet but not for the ferromagnetic case it may be suggested that perhaps the high temperature expansions have not actually given us much useful information about the Heisenberg ferromagnet after all. These problems should be resolved before we consider the phase transition in the quantum Heisenberg model as fully understood.

9.4

Statistical mechanics versus quantum field theory

At the end of chapter 1 we briefly commented that the path integral formulation of quantum field theory where averages of operators A are calculated as 1 A = [dφ]AeS/¯h (9.74) Z

with Z=

[dφ]eS/¯h

(9.75)

where S is the action looks “formally” equivalent to the statistical mechanical formulas where averages of operators O are calculated as 1 Oe−E/kB T (9.76) O = Z all states

where the partition function Z is

Z=

e−E/kRT

(9.77)

all states

where E is the interaction energy of the system. This formal equivalence is fully developed in the discipline of lattice gauge theory and it is out of place and beyond the scope of this book to develop this in detail. Nevertheless it is perhaps not out of place to conclude this chapter by giving what can be called a dictionary which translates the language from one field to the other. Consider the Euclidean field theory of the n component nonlinear sigma model in two dimensions where the action is 2 2 n ∂φj (x) S= (9.78) ∂xµ j=1 µ=1 and the n component field φj (x) satisfies the constraint n

(φj (x))2 = 1.

(9.79)

j=1

For this field theory the path integral may be precisely defined by “imposing a lattice cut-off” and writing derivatives as differences. When this is done we obtain the interaction energy of the n vector model which for n = 1 is the Ising model and for n ≥ 2

¾


is studied by high temperature expansions. The field theory of the nonlinear sigma model in continuum space is then obtained by taking the scaling limit as discussed in chapter 5. With this as the definition of the Euclidean quantum field theory of the nonlinear sigma model we may now make contact between the language of statistical mechanics and the language of quantum field theory. Some of the important elements in the translation of the two languages are presented in Table 9.26. Table 9.26 A dictionary of translation of classical statistical mechanics of the n vector model and the quantum field theory of the nonlinear sigma model.

Statistical mechanics Classical interaction energy Sum over all states Scaling limit Temperature T Tc > 0 Tc = 0 Kosterlitz–Thouless phase

Quantum field theory Lagrangian Lattice regularized path integral Renormalization procedure h ¯ Nontrivial fixed point Asymptotically free Quantum electrodynamics

The first important translation in Table 9.26 is that ¯h in quantum field theory is equivalent to kB T in statistical mechanics. This may at first sight seem very strange because in quantum field theory units are often said to be chosen such that h ¯ is set equal to unity and this merely sets a scale whereas the major goal in the study of statistical mechanics is to determine the properties of the system as a function of T It is thus most important to recognize that in fact for the nonlinear sigma model h ¯ must not be treated as a scale setting number but should be thought of as a “coupling constant” which can in principle be varied. For the n vector model in two dimensions there are three distinct cases for n = 1, n = 2, and n ≥ 3. Consider first n = 1. This is the Ising model where there is a critical point at T = Tc > 0 and in the vicinity of that critical point the scaling limit exists. In the corresponding field theory language the value of h ¯ or if you will of the coupling constant g which corresponds to Tc is called a nontrivial fixed point. We have seen that there are two distinct scaling limits which may be constructed by approaching Tc either from above or below. This same two-phase structure must therefore exist for the n = 1 component field theory. The second case to consider is n = 2. In this case the n vector model has a Kosterlitz–Thouless temperature TKT > 0 such that for T < TKT the system has no mass gap. Thus for this system there are also two ways to compute a scaling limit. Either T → Tc +, in which case there will be massive excitations in the system, or T can be allowed to have any value below TKT , in which case the excitations are massless. The nonlinear sigma model will therefore also have these two different phases and we note that the low temperature phase is in the same spirit as quantum electrodynamics where the photon in massless but the coupling constant can be varied. The final case is n ≥ 3. In this case the n vector model becomes massless only at

Appendix: The expansion coefficients for the susceptibility on the square lattice

¾

T = 0 and the only phase which can be constructed is the high temperature phase T → 0+. In quantum field theory this is called “asymptotic freedom”. We saw that this high temperature phase had the unfortunate feature that it could not be studied in the limit T → 0 by means of high temperature series expansions because the leading singularities were not at T = 0 but were in the complex plane with a positive real part. We now come to what is quite possibly a controversial aspect to the comparison of quantum field theory and statistical mechanics because quantum field theories are very often defined by “dimensional regularization” which in the present case amount to allowing the dimension D of the system to be D > 2. At least in some formal sense this can be done and in the same formal sense the theorem proven in chapter 5 on lack of order for the n ≥ 2 vector model breaks down and there will be a nontrivial fixed point for Tc > 0. Thus for D > 2 a low temperature phase of the nonlinear sigma model can be constructed and then the dimension D may be allowed to return to two. However, any such results constructed in this manner will have no analogue or relevance to the n vector model constructed on the lattice where only a high temperature phase exists.

9.5


We list here the 21 coefficients ak (n) for the susceptibility on the square lattice as given in [6, pages 15838–15840] a1 (n) = 4

(9.80)

a2 (n) = 12 a3 (n) = (72 + 32n)/(n + 2)

(9.81) (9.82)

a4 (n) = (200 + 76n)/(n + 2) a5 (n) = 8(284 + 147n + 20n2 )/((n + 2)(n + 4))

(9.83) (9.84)

a6 (n) = 16(780 + 719n + 201n2 + 19n3 )/((n + 2)2 (n + 4))

(9.85)

For the remaining coefficients we write ak (n) = Pk (n)/Qk (n)

(9.86)

and list Pk (n) and Qk (n) separately. P7 (n) = 16(26064 + 38076n + 20742n2 + 5280n3 + 655n4 + 32n5 ) 3

Q7 (n) = (n + 2) (n + 4)(n + 6)

(9.87) (9.88)

P8 (n) = 4(283968 + 383568n + 186912n2 + 41000n3 + 4392n4 + 187n5) (9.89) Q8 (n) = (n + 2)3 (n + 4)(n + 4)(n + 6)

(9.90)

¾


P9 (n) = 8(3123456 + 4186336n + 2087128n2 + 492220n3 + 62386n4 +4161n5 + 116n6 ) 3

(9.91)

Q9 (n) = (n + 2) (n + 4)(n + 6)(n + 8)

(9.92)

P10 (n) = 16(33868800 + 66758016n + 53214272n2 + 22126648n3 +5211372n4 + 719330n5 + 58789n6 + 2684n7 + 55n8 )

(9.93)

4

2

Q10 (n) = (n + 2) (n + 4) (n + 6)(n + 8)

(9.94)

P11 (n) = 32(3695370240 + 9913385984n + 11437289216n2 + 7427564992n3 +2989987696n4 + 776848144n5 + 132130072n6 + 14693596n7 + 1052911n8 +46923n9 + 1225n10 + 16n11 ) Q11 (n) = (n + 2)5 (n + 4)3 (n + 6)(n + 8)(n + 10)

(9.95) (9.96)

P12 (n) = 16(4990955520 + 11511967232n + 10992991488n2 + 5609888352n3 +1649559472n4 + 281912408n5 + 27080244n6 + 1334568n7 + 22368n8 −199n9 + 5n10 ) Q12 (n) = (n + 2)5 (n + 4)2 (n + 6)(n + 8)(n + 10)

(9.97)

P13 (n) = 64(162478080000 + 406158981120n + 431982472192n2 +25491324928n3 + 90288340864n4 + 19721001832n5 + 2561904944n6 +170376718n7 + 1211742n8 − 616479n9 − 37625n10 − 635n11 + 4n12 ) 5

3

Q13 (n) = (n + 2) (n + 4) (n + 6)(n + 8)(n + 10)(n + 12)

(9.98) (9.99)

P14 (n) = 16(21007560867840 + 63770201063424n + 84400316350464n2 63787725946880n3 + 30245054013440n4 + 9275137432448n5 1810742519232n6 + 204284290016n7 + 7651128688n8 − 1135009056n9 −177417600n10 − 9861216n11 − 161554n12 + 5277n13 + 172n14)

(9.100)

Q14 (n) = (n + 2)6 (n + 4)3 (n + 6)2 (n + 8)(n + 10)(n + 12)

(9.101)


¾

P15 (n) = 16(9537349044142080 + 3464137448513888n +56169030631292928n2 + 53537028436525056n3 +33216830381735936n4 + 14006542675035136n5 +777531907925504n7 + 87227510881024n8 + 1944102682560n9 −1061882170400n10 − 183809104832n11 − 14563826832n12 −515944376n13 + 583019n14 + 1259012n15 + 43647n16 + 512n17 ) Q15 (n) = (n + 2)7 (n + 4)3 (n + 6)3 (n + 8)(n + 10)(n + 12)(n + 14)

(9.102) (9.103)

P16 (n) = 4(410123375487221760 + 1537129944780374016n + 258398341169047142n2 +2566975595695570944n3 + 1669084283351334912n4 + 741114014711103488n5 +225948162044579840n6 + 45385102417264640n7 + 4996850176026624n8 −6480424496896n9 − 122658733213440n10 − 20909429640960n11 −1752208241536n12 − 56642417728n13 + 3062606512n14 +412508368n15 + 18713696n16 + 395328n17 + 3083n18) 7

4

3

Q16 (n) = (n + 2) (n + 4) (n + 6) (n + 8)(n + 10)(n + 12)(n + 14)

(9.104) (9.105)

P17 (n) = 8(35361815028050165760 + 138539666887258669056n +245436909326998437888n2 + 259375081142913859584n3 +181396134616565809152n4 + 87793764370648399872n5 +29673202500166647808n6 + 6770762601709142016n7 + 893508862130341888n8 +5229394076767232n9 − 24934992828139008n10 − 5547918408527104n11 −625740097598720n12 − 33521607263744n13 + 738107699392n14 +272358030048n15 + 21867513640n16 + 937447020n17 + 22261658n18 +250495n19 + 692n20 ) 7

5

3

Q17 (n) = (n + 2) (n + 2) (n + 6) (n + 8)(n + 10)(n + 12)(n + 14)(n + 16)

(9.106) (9.107)

P18 (n) = 16(758936838540424642560 + 3343934774878956158976n +6720650800795024883712n2 + 8141819950133007089664n3 +6611686534391523180544n4 + 3777832155850593533952n5 +1543669324445646061568n6 + 443919373830353158144n7 +82653304109539049472n8 + 6333077582288386048n9 −1419388196952978432n10 − 578178922758906368n11 − 99290612095487744n12 −9300735775467264n13 − 256677768200576n14 + 53852331942080n15 +8516631212960n16 + 629479458104n17 + 26896421724n18 +607483694n19 + 2912825n20 − 154080n21 − 2313n22)

(9.108)

Q18 (n) = (n + 2)8 (n + 4)5 (n + 6)3 (n + 8)2 (n + 10)(n + 12)(n + 14)(n + 16) (9.109)

¾


P19 (n) = 32(293792962132985669222400 + 1452203904509587992084480n +3305764874023051160715264n2 + 4587498272216547279765504n3 +4326547244550747303444480n4 + 2922303204727243671601152n5 +1446836388063996470624256n6 + 524786164553898279829504n7 +134502578329442459254784n8 + 2110487872734821885952n9 +411991601001072488448n10 − 794587338452494176256n11 −242141294836583751680n12 − 39373307992978213888n13 −3583854665917282560n14 − 51879072941552128n15 +36069375006840576n16 + 5868286096676352n17 +508264525824336n18 + 27200961065872n19 + 811699909040n20 +3015005636n21 − 793163459n22 − 32254806n23 − 562185n24 − 3824n25) Q19 (n) = (n + 2)9 (n + 4)5 (n + 6)3 (n + 8)3 (n + 10)(n + 12)

(9.110)

(n + 14)(n + 16)(n + 18)

(9.111)

P20 (n) = 25184031413058833177640960 + 120991848351738482367922176n +266735758564462825159262208n2 + 356790558744797070187560960n3 +322230995604339689974136832n4 + 206410863077042429645291520n5 +95396352174319203631759360n6 + 31338576528595789665009664n7 +6741729364322236678275072n8 + 606442638553174662709248n9 −158967889827748034248704n10 − 78160816104265974611968n11 −16419585036974248984576n12 − 1908913019215816540160n13 −62769211853172834304n14 + 19736004882625224704n15 +4130688305419677696n16 + 421867767284303872n17 + 25241612021992960n18 +693193236915968n19 − 18814089206912n20 − 2684704080320n21 −120121949760n22 − 2872757568n23 − 35919232n24 − 178096n25 9

5

3

(9.112)

3

Q20 (n) = (n + 2) (n + 4) (n + 6) (n + 8) (n + 10)(n + 12)(n + 14) ×(n + 16)(n + 18)

(9.113)


¾

P21 (n) = 1351534773860942603511398400 + 6365460757030282723495772160n +13733120620155454487896522752n2 + 17929816694573858749176348672n3 +15741441712440400461107822592n4 + 9736866800557416285986619392n5 +4291917210346656516848746496n6 + 1307842644179764469724872704n7 +238056916142751738992001024n8 + 3543432139088928241090560n9 −12741108714260109576372224n10 − 4295859766813135292465152n11 −755125851031606609051648n12 − 65023497347980126945280n13 +2821567036764628910080n14 + 1727752072630205923328n15 +264572837767576051712n16 + 22840865368902557696n17 +1035703381014509568n18 − 6131507476388352n19 −4493143518510080n20 − 338088589058432n21 −14045533700352n22 − 35536140880n23 −5200818400n24 − 35916176n25 − 47360n26 Q21 (n) = (n + 2)9 (n + 4)5 (n + 6)3 (n + 8)3 (n + 10)(n + 12)(n + 14)

(9.114)

(n + 16)(n + 18)(n + 20)

(9.115)

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References

¾

[16] M.F. Sykes, D.S. Gaunt, J.D. Roberts and J.A. Wyles, High temperature series for the susceptibility of the Ising model II. Three dimensional lattices, J. Phys. A 5 (1972) 640–652. [17] D.S. Gaunt and M.F. Sykes, The critical exponent γ for the three dimensional Ising model, J. Phys. A12 (1979) L25–L28 [18] S. McKenzie, High-temperature reduced susceptibility of the Ising model, J. Phys. A 8 (1975) L102–L105 [19] P. Butera and M. Comi, Critical specific heats of the N vector spin models on the simple cubic and bcc lattices, Phys. Rev. B 60 (1999) 6749–6760. [20] M.F. Sykes, D.L. Hunter, D.S. McKenzie and R. Heap, J. Phys. A 5 (1972) 667– 673. [21] A.J. Guttmann and I.G. Enting, Series studies of the Potts model:I. the simple cubic Ising model, J. Phys. A 26 (1993) 807–821. [22] A.J. Guttmann and I.G. Enting, The high–temperature specific heat exponent of the 3D Ising model, J. Phys. A 27 (1994) 8007–8010. [23] P.S. English, D.L. Hunter and C. Domb, Extension of the high-temperature, free energy series for the classical vector model of ferromagnetism in general spin dimensionality, J. Phys. A 12 (1979) 2111–2130. [24] M. Kosterlitz and D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C6 (1973) 1181–1203. [25] A.J. Guttmann, On the critical behavior of self-avoiding walks, J. Math. Phys. A 20 (1987) 1839–1854. [26] A.J. Guttmann, The high-temperature susceptibility and spin-spin correlation function of the three dimensional Ising model, J.Phys. A20 (1987u) 1855–1863. [27] A.J. Guttmann, Asymptotic analysis of power-series expansions in Phase Transition and Critical Phenomena, (Academic Press, San Diego 1989) ed. C. Domb and J.L.lebowitz, vol. 13. pp. 1–234. [28] G.A. Baker, Jr., H.E. Gilbert, J. Eve and G.S. Rushbrooke, Phys. Rev. 164 (1967) 800–817. [29] G.S. Rushbrooke and P.J. Wood, On the Curie points and high temperature susceptibilities of Heisenberg model ferromagnetics, Mol. Phys. 1 (1958) 257–283. [30] R.L. Stephenson, K. Pirnie, P.J. Wood and J. Eve, On the high temperature susceptibility and specific heat of the Heisenberg magnet for a general spin, Phys. Letts. A27 (1968) 2–3. [31] K–K. Pan, Néel temperature of quantum quasi-two-dimensional Heisenberg antiferromagnets, Phys. Letts. A 271 (2000) 291–295. [32] G.S. Rushbrooke and P.J. Wood, On the high temperature staggered susceptibility of Heisenberg model antiferromagnetics, Mol. Phys. 6 (1963) 409–421. [33] J. Oitmaa and E. Bornilla, High temperature study of the spin 1/2 Heisenberg ferromagnet, Phys. Rev. B 53 (1996) 14228–14235. [34] J. Oitmaa and W-H. Zheng, Curie and Néel temperatures of quantum magnets, J. Phys. Cond. Mat. 16 (2004) 8653–8660.


Part III Exactly Solvable Models

A thing of beauty is a joy forever. John Keats


10 The Ising model in two dimensions: summary of results The remainder of this book will be devoted to statistical models for which exact computations may be carried out. These are called integrable or solvable models. By far the most important and most extensively studied of these systems is the Ising model in two dimensions defined by the interaction energy E=−

Lh Lv

{E h σj,k σj,k+1 + E v σj,k σj+1,k + Hσj,k }

(10.1)

j=1 k=1

where the variables σj,k = ±1 are located at the j row and k column of a square lattice with Lv rows and Lh columns, and either free, cylindrical or periodic (toroidal) boundary conditions may be imposed. This interaction energy (10.1) was first considered by Lenz [1] in 1920 and the free energy in one dimension was computed by Ising [2] in 1925. In two dimensions the existence of long range order was proven by Peierls [3] in 1936 and the critical temperature was located by Kramers and Wannier [4, 5] in 1941 by means of a duality transformation. The discovery that exact computations can be done for the two-dimensional Ising model (10.1) at H = 0 was made in 1944 by Lars Onsager [6] who exactly computed the free energy. Since then many properties including the spontaneous magnetization and correlation functions have been exactly computed and the magnetic susceptibility has been studied in substantial detail. An historical overview of some of the major developments is given in Table 10.1. The great importance of the Ising model in two dimensions is that at H = 0 it exhibits a phase transition at a critical temperature T = Tc . The model exhibits all the features of critical phenomena presented in chapter 5 and in particular the critical exponents at H = 0 which are given in Table 10.2 have been computed exactly and are seen to obey the scaling laws of chapter 5. Indeed many of the scaling laws of chapter 5 were first seen in the Ising model and it is no exaggeration to say that the general theory of critical phenomena and scaling theory presented in the previous chapters is a generalization of results first obtained for the Ising model. This general theory is universally used to describe critical phenomena in three dimensions. The first purpose of this chapter is to summarize the many exact results of the Ising model at H = 0 on which the general theory of critical phenomena is erected. We will also present the exact results for surface phenomena which also exhibit critical exponents and scaling behavior. In addition we will give the results for layered systems

¾

The Ising model in two dimensions: summary of results

Table 10.1 Historical overview of major developments in the study of the two dimensional Ising model at H = 0.

Date 1920 1925 1936 1941 1944 1949 1949 1949

Author(s) Lenz [1] Ising [2] Peierls [3] Kramers, Wannier [4, 5] Onsager [6] Onsager [7] Kaufman [8] Kaufman, Onsager [9]

1952 1963 1963 1966-67

Yang [10] Kastelyn [11] Montroll, Potts, Ward [12] Cheng, Wu [13, 14]

1967 1968-69 1969 1973-1976

McCoy, Wu [15] McCoy, Wu [16–18] Griffiths [19] Barouch, McCoy, Tracy, Wu [20–22] Sato, Miwa, Jimbo [23–27] McCoy , Tracy, Wu [28–30] McCoy, Perk, Wu [31–34]

1976-1980 1978 1980-1981 1981 1999-2001 2004 2007

2007

Jimbo,Miwa [35] Orrick, Nickel, Guttmann, Perk [36–39] Zenine, Boukraa, Hassani, Maillard [41–43] Boukraa, Hassani Maillard, McCoy Orrick, Zenine [44] Boukraa, Hassani Maillard, McCoy Weil, Zenine [45]

Property Ising model introduced On dimensional model solved Existence of spontaneous magnetization Duality and Tc Free energy Spontaneous magnetization Partition function on torus Short range correlations two-spin correlation at Tc Spontaneous magnetization Ising model reduced to dimers Correlations as determinants Large separation behavior of the two-point function Boundary critical behavior Random layered lattices Random lattice singularities Painlevé III representation for the scaled two point function Holonomic field theory n point functions Partial difference equations for correlation functions Painlevé VI for diagonal correlations Bulk susceptibility Fuchsian equation for 3 and 4 particle contributions to bulk susceptibility Factorization of form factors

Diagonal Ising susceptibility

where the interaction energies E v and E h vary randomly from row to row and show that, for these random systems, the critical exponent description does not hold. However, all of the results at H = 0 depend on very special properties which allow the exact solvability and these remarkable properties can not possibly be shared by any real system in the laboratory. It may therefore be asked why the phenomenology derived from this Ising model seems to be so very accurate and powerful when applied


¾

Table 10.2 Critical exponents for the two-dimensional Ising model.

Property

Critical exponent Bulk properties

Specific heat at α = 0 ln |T − Tc | Spontaneous magnetization β = 1/8 Magnetic susceptibility γ = 7/4 Magnetization at T = Tc δ = 1/15 Correlation length ν=1 Anomalous dimension η = 1/4 Boundary properties Specific heat αb = 1 Spontaneous magnetization βb = 1/2 Magnetic susceptibility γb = 0 ln |T − Tc | Magnetization at Tc δb = 1 Hb ln |Hb | Correlation length νb = 1 Anomalous dimension at Hb = 0 ηb = 1 Anomalous dimension at Hb = 0 ηb = 4 to real systems in three dimensions and what, if anything, is missing in the intuition derived from the exact results obtained for the Ising model. In particular since the general theory of critical phenomena was erected as a generalization of the Ising model and confirmed by all other models which have special properties (to be discussed in later chapters) which allow them to be exactly solved, we may ask if it could happen that this general theory of critical phenomena is only applicable to these special integrable systems. We attempt to give some insight into this most important question by considering the Ising model at H = 0. An historical overview of some of the major developments is given in Table 10.3. For small H a perturbation technique [48] shows that in the scaling limit there is a sort of “confinement phenomenon” which takes place and a very remarkable computation done in 1989 by Zamolodchikov [50] shows that, in the scaling limit at T = Tc , the model is again integrable. In the present chapter we will summarize and discuss these results. In the following chapter we will present in detail one of the methods of computing the partition function Table 10.3 Historical overview of major developments in the study of the two dimensional Ising model at H = 0.

Date 1952 1952,1967 1978 1984 1988 2003

Author(s) Lee, Yang [46] Lee, Yang [46] McCoy, Wu [47] McCoy, Wu [48] Isakov [49] Zamolodchikov [50] Fonseca, Zamolodchikov [51]

Property Circle theorem Solution at H/kB T = iπ/2 Confinement for T < Tc Nonanalyticity at H = 0 for T < Tc E8 scaled Ising model at T = Tc Extended analyticity for free energy

¾


on the finite lattice and derive determinental expressions for the correlation functions. In chapter 12 we derive the spontaneous magnetization and the form factor expansion of the correlation functions from the determinants. For all other derivations we refer the reader to the original derivations in the literature or for some of the older results to the book The Two Dimensional Ising Model by T.T. Wu and the present author [52].

10.1

The homogeneous lattice at H = 0

We begin by presenting the results for the homogeneous Ising model (10.1) on a lattice with Lv rows and Lh columns. 10.1.1

Partition function on the torus

For periodic boundary conditions, the partition function was found by Kaufman [8] to be given by the sum of four terms Z(T, Lv , Lh ) 1 = {−Zee (T, Lv , Lh ) + Zeo (T, Lv , Lh ) + Zoe (T, Lv , Lh ) + Zoo (T, Lv , Lh )}(10.2) 2 where Zj (T, Lv , Lh ) = 2Lv Lh [cosh 2K h cosh 2K v − sinh 2K h cos θ1 − sinh 2K v cos θ2 ]1/2(10.3) θ1

θ2

where K h = E h /kB T and K v = E v /kB T

(10.4)

and θ1 and θ2 are chosen as in Table 10.4 with n1 = 1, 2, · · · , Lh and n2 = 1, 2, · · · , Lv . Table 10.4 Allowed values of θi

j ee eo oe oo

θ1 2πn1 /Lh 2πn1 /Lh π(2n1 − 1)/Lh π(2n1 − 1)/Lh

θ2 2πn2 /Lv π(2n2 − 1)/Lv 2πn2 /Lv π(2n2 − 1)/Lv

The square root in (10.3) is only apparent because all terms appear in pairs. These apparent square roots are defined to be positive when T > Tc . When Lv and Lh are both even we explicitly have

Lh /2 Lv /2

Zoo = 2

Lv Lh

{cosh 2K v cosh 2K h − sinh 2K h cos π(2n1 − 1)/Lh

n1 =1 n2 =1

− sinh 2K v cos π(2n2 − 1)/Lv }2 ,

(10.5)


¾

Lv /2

Zeo = 2Lv Lh

{([cosh 2K v cosh 2K h − sinh 2K v cos π(2n2 − 1)/Lv ]2 − sinh2 2K h )

n2 =1

Lh /2−1

×

[cosh 2K v cosh 2K h − sinh 2K h cos 2πn1 /Lh − sinh 2K v cos π(2n1 − 1)/Lv ]2 }

n1 =1

(10.6)

Lh /2

Zoe = 2Lv Lh

{([cosh 2K v cosh 2K h − sinh 2K h cos π(2n1 − 1)/Lh ]2 − sinh2 2K v )

n1 =1

Lh /2−1

×

[cosh 2K v cosh 2K h − sinh 2K v cos 2πn2 /Lv − sinh 2K h cos π(2n2 − 1)/Lh ]2 },

n2 =1

(10.7) and Zee = 2Lv Lh (1 − sinh2 2K h sinh2 2K v )

Lh /2−1

×

([cosh 2K v cosh 2K h − sinh 2K h cos π2n1 /Lh ]2 − sinh2 2K v )

n1 =1

Lv /2−1

×

([cosh 2K v cosh 2K h − sinh 2K v cos π2n2 /Lv ]2 − sinh2 2K h )

n2 =1

n1 =1

n2 =1

Lh /2−1 Lv /2−1

×

{cosh 2K h cosh 2K v − sinh 2K h cos 2πn1 /Lh − sinh 2K v cos 2πn2 /Lv }2 , (10.8)

10.1.2

Zeros of the partition function

The zeroes in the complex T plane of the four Zj (T, Lv , Lh ) are obvious from the factored form (10.3) but the zeros of the partition function itself are tedious to locate. However, for large Lv and Lh the limiting distributions of the zeroes of the partition function and of the four Zj (T, Lv , Lh ) are identical and are obtained from the equation cosh 2K h cosh 2K v − sinh 2K h cos θ1 − sinh 2K v cos θ2 = 0

(10.9)

with 0 ≤ θ1 , θ2 ≤ 2π. In the isotropic case when E v = E h = E the formula for the location of the zeros (10.9) reduces to 1 + sinh2 2K − sinh 2K(cos θ1 + cos θ2 ) = 0 from which we see that

(10.10)

¾


1 {cos θ1 + cos θ2 ± i[4 − (cos θ1 + cos θ2 )2 ]1/2 ]} 2 and thus the zeros lie on a circle sinh 2K =

| sinh 2K|2 = 1.

(10.11)

(10.12)

For the general anisotropic case the zeros obtained from (10.9) fill up areas. However for real E v and E h these regions of zeros pinch the real axis only at points Tc determined from sinh 2E v /kB Tc sinh 2E h /kB Tc = ±1. (10.13) Consequently these zeros satisfy the condition discussed in chapter 5 which is required for a second order phase transition. It is convenient in the isotropic case to study the zeros of the partition function by introducing the variable x = e−2K (10.14) and write ˜ Z(T, Lv , Lh ) = xLv Lh Z(x)

(10.15) ˜ where Z(x) is a polynomial in x of degree Lv Lh given by (10.2) with Zj (T, Lv , Lh ) → Z˜j (x) where Z˜j (x) = {(1 + x2 )2 + 2x(1 − x2 )(cos θ1 + cos θ2 )}1/2 (10.16) θ1

θ2

which is a polynomial in u = x2 . The zeros of the polynomials Z˜j (x) lie on the two circles √ |x ± 1| = 2

(10.17)

(10.18)

but the zeros of the full partition function lies on these circles only in the thermodynamic limit. It was discovered in 1974 by Brascamp and Kunz [53] that for the isotropic lattice the zeros of the partition function will lie exactly on the circles (10.18) if Lh is even and the boundary conditions are chosen as follows: 1) There are periodic boundary conditions in the horizontal direction. 2) The spins in the upper boundary row interact with a row of spins all fixed to be +. 3) The spins in the lower boundary row interact with a row of spins which are fixed and alternate + and −. For this choice of boundary conditions the polynomial partition function normalized to unity at x = 0 is ZBK (x) =

Lh /2 Lv

{(1 + x2 )2 + 2x(1 − x2 )[cos((2j − 1)π/Lh ) + cos(kπ/(Lv + 1))]}.

j=1 k=1

(10.19) We compare the zero distributions for the toroidal and the Brascamp–Kunz boundary conditions by plotting the zeros for an 18 × 18 lattice in the complex u plane in


¾

Fig. 10.1 Comparison of the zeros in the plane u = x2 of the Ising partition functions with periodic boundary conditions on the left and Brascamp–Kunz boundary conditions on the right for an 18 × 18 lattice at H = 0.

Fig. 10.1. Zeros for various finite size lattices with other boundary conditions have been studied by Matveev and Shrock [54]. 10.1.3

Bulk free energy per site

The free energy per site is computed from the partition function (10.2) as −F/kB T =

lim

Lv ,Lh →∞

1 lnZ(T, Lv , Lh ) Lv Lh

(10.20)

and when T is in a zero-free region of the plane we obtain the famous result of Onsager [6]: F/kB T = −ln2 2π 2π 1 − 2 dθ1 dθ2 ln[cosh 2K h cosh 2K v − sinh 2K h cos θ1 − sinh 2K v cos θ2 ] 8π 0 0 (10.21) This free energy is independent of the signs of E v and E h and we will henceforth take these two interaction energies to be real and positive. The argument of the logarithm in (10.21) is nonnegative for 0 ≤ θ1 , θ2 < 2π. The argument only vanishes at θ1 , θ2 = 0, π and this vanishing imposes the restriction T = Tc cosh 2E h /kB Tc cosh 2E v /kB Tc ± sinh 2E h /kB Tc ± sinh 2E v /kB Tc = 0

(10.22)

which is equivalent to (10.13) which is the location in the complex T plane where the zeros of the partition function pinch the real axis. We will by convention define Tc to

¾


be the solution of (10.13) with the positive sign on the right-hand side. We also note v that, if we solve (10.13) (with the positive sign) for e2E /kB Tc , we obtain an equivalent condition for Tc of h e2E /kB Tc + 1 2E v /kB Tc e (10.23) = 2E h /k T B c − 1 e and the companion equation with E v ↔ E h . The free energy (10.21) is singular at T = Tc . However, this is the only feature of the distribution of the zeros of the partition function that remains in the free energy. In particular the free energy may be analytically continued through and beyond the regions in the complex plane where the finite size partition function has zeros. The internal energy is obtained from the free energy (10.21) as u=

∂βF = −E h σ0,0 σ0,1 − E v σ0,0 σ1,0 ∂β

(10.24)

where β = 1/kB T and 1 σ0,0 σ0,1 = 2π with

2π

0

(1 − α1 eiθ )(1 − α2 e−iθ ) dθ (1 − α1 e−iθ )(1 − α2 eiθ )

1/2

α1 = e−2K tanh K h and α2 = e−2K coth K h v

v

(10.25)

(10.26)

and σ0,0 σ0,1 = σ0,0 σ1,0 with K and K interchanged. The square root is defined positive at θ = π. In the isotropic case E v = E h = E the internal energy may be written in terms of the complete elliptic integral h

π/2

K(k) = 0

dφ = (1 − k 2 sin2 φ)1/2

v

1 0

dx 2[x(1 − x)(1 − k 2 x)]1/2

(10.27)

as u = −E coth 2βE[1 + 2π −1 (2 − tanh2 2βE)K(k)] = −E{2[1 + (1 − k 2 )1/2 ]}1/2 {1 ∓ π −1 (1 − k 2 )1/2 K(k)} where k=2

2 sinh 2βE = 2 (cosh 2βE) sinh 2βE + sinh−1 2βE

(10.28)

(10.29)

and in the second line of (10.28) the minus (plus) sign is chosen for T > Tc (T < Tc ). We note for real T that |α1 | < 1 but that α2 has no such restriction. In particular we may have α2 = 1 and at this point the integral in (10.25) fails to be analytic because the square root branch cuts at α±1 pinch the contour of integration. From 2 (10.26) we see that the condition α2 = 1 is identical to the form of the Tc condition (10.23). We also note that near Tc we have


α2 ∼ 1 − (β − βc )2{E v +

¾

Eh } sinh 2E h /kB Tc

= 1 − (β − βc )2{E v + E h sinh 2E v /kB Tc } = 1 − (β − βc )2 tanh 2E v /kB Tc {E v coth 2E v /kB Tc + E h coth 2E h /kB Tc } (10.30) where in the last line we have used the identity coth 2E h /kB Tc = coth 2E v /kB Tc

sinh 2E v /kB Tc sinh 2E h /kB Tc

1/2 =

cosh 2E v /kB Tc cosh 2E h /kB Tc

(10.31)

which follows from the condition for Tc (10.13) or (10.23). Near Tc we find 2 (β − βc )(E v + E h sinh 2E v /kB Tc ) ln |β − βc | π (10.32) where gdx = arctan sinhx is the Gudermannian of x. From the internal energy (10.24) we obtain the specific heat and find that near Tc σ0,0 σ0,1 ∼ 2π coth 2Kch gd2Kch −

c=

∂u 2kB ∼− (Kch2 sinh 2Kcv + 2Kcv Kch + Kcv2 sinh 2Kch )ln|1 − T /Tc|+ O(1) (10.33) ∂T π

where the term O(1) is the same for T above and below Tc . Thus the specific heat has a logarithmic divergence at Tc which corresponds to a critical index of α = 0. The specific heat is plotted in Fig. 10.2 as a function of temperature for various value of the anisotropy. 1.5

= Eh/Ev = 1 = Eh/Ev = 0.1 = Eh/Ev = 0

1.0 c/k

0.5

0

1.0

2.0 2kBT/Eh+Ev

3.0

Fig. 10.2 Specific heat of the Ising model for E v /E h = 1, E v /E h = 0.1 and E v /E h = 0.

¾


10.1.4

Partition function at T = Tc

If T = Tc we see from (10.2) and (10.21) that for Lv , Lh 1 Z(T, Lv , Lh ) = e−Lv Lh F/kB T {1 + terms exponentially small in Lv Lh }}.

(10.34)

However, if T = Tc (10.34) does not hold and instead we have the result of Ferdinand and Fisher [60] that if Lv , Lh 1 then

θ3 (0; τ ) θ4 (0; τ ) 1 θ2 (0; τ ) |+| |+| | exp(−Lv Lh F/kB Tc ) (10.35) | Z(Tc , Lv , Lh ) ∼ 2 η(τ ) η(τ ) η(τ ) where η(τ ) = eiτ /24

∞

(1 − eijτ )

(10.36)

j=1

and the theta functions are θj (v; τ ) θ1 (v; τ ) = θ2 (v; τ ) = θ3 (v; τ ) = θ4 (v; τ ) =

∞

2

(−1)n eiτ (n+1/2) e2πiv(n+1/2)/K

n=−∞ ∞ n=−∞ ∞

2

eiτ (n+1/2) e2πiv(n+1/2)/K 2

eiτ n e2πivn/K

n=−∞ ∞

2

(−1)n eiτ n e2πivn/K

(10.37) (10.38) (10.39) (10.40)

n=−∞

where τ =i 10.1.5

Lv cosh 2Kch . Lh cosh 2Kcv

(10.41)

Spontaneous magnetization

The spontaneous magnetization M− (T ) was announced by Onsager [6] in 1949 and a derivation was given by Yang [10] in 1952. The remarkably simple result is M− (T ) = [1 − (sinh 2E h /kB T sinh 2E v /kB T )−2 ]1/8 for 0 < T ≤ Tc = 0 for T > Tc . (10.42) This shares the feature with the free energy that it is singular only at the points where the Tc condition (10.13) holds and that there are no further singularities at the location of the zeros of the partition function. Near Tc we have sinh 2E h /kB T sinh 2E v /kB T ( ' ∼ 1 + (β − βc ) 2E h coth 2E h /kB Tc + 2E v coth 2E v /kB Tc

(10.43)

and thus from (10.42) we have as T → Tc − 0 11/8 M− (T ) ∼ (β − βc )4(E h coth 2E h /kB Tc + E v coth 2E v /kB Tc )

(10.44)

from which we see that the critical exponent β = 1/8.


¾

The spontaneous magnetization is plotted in Fig. 10.3 for the isotropic case E v = E along with the magnetization in the boundary row in the half plane lattice given in (10.259). h

Fig. 10.3 Comparison of the spontaneous magnetization M (10.42) with the magnetization M1 (10.259) in the boundary row of the half plane lattice for the isotropic lattice E v = E h as a function of temperature.

10.1.6

Row and diagonal spin correlation functions

It was shown by Montroll, Potts and Ward [12] that all spin correlation functions C(M, N ) = σ0,0 σM,N may be expressed as determinants. Indeed, every correlation function may be expressed as a determinant in an infinite number of ways in the thermodynamic limit. However, more is known about the diagonal and row correlation than is known for the case of general C(M, N ) and therefore we will treat these cases separately. The diagonal correlation C(N, N ) = σ0,0 σN,N and the row C(0, N ) = σ0,0 σ0,N correlation functions can both be written as N × N Toeplitz determinants

DN

a0 a1 = . ..

a−1 a0 .. .

· · · a−N +1 · · · a−N +2 .. .

(10.45)

aN −1 aN −2 · · · a0 with an =

1 2π

2π

dθe−inθ

0

where for C(N, N ) α1 = 0, and for C(0, N )

(1 − α1 eiθ )(1 − α2 e−iθ ) (1 − α1 e−iθ )(1 − α2 eiθ )

α2 = (sinh 2K v sinh 2K h )−1

1/2 (10.46)

(10.47)

¾


α1 = e−2K tanh K h , v

α2 = e−2K coth K h v

(10.48)

and the square roots are defined to be positive at θ = π. These determinants are very efficient for the calculation of the correlations when N is small. For the diagonal correlation function all matrix elements may be expressed in terms of the integral representation the hypergeometric function [55, chapter 2] valid for Re c > Re b > 0 : 1 Γ(c) F (a, b; c; t) = dxxb−1 (1 − x)c−b−1 (1 − tx)−a . (10.49) Γ(c − b)Γ(b) 0 For T < Tc with

t = (sinh 2K v sinh 2K h )−2 = α22 < 1

(10.50)

we have for n ≥ 0 Γ(n + 1/2) F (−1/2, n + 1/2; n + 1; t) π 1/2 n!

(10.51)

Γ(|n| + 1/2) F (1/2, |n| + 1/2; |n| + 2; t). 2π 1/2 (|n| + 1)!

(10.52)

an = tn/2 and for n ≤ −1 an = t|n|/2 For T > Tc with

t = (sinh 2K v sinh 2K h )2 = α−2 2 < 1

(10.53)

t(n+1)/2 Γ(n + 1/2) F (1/2, n + 1/2; n + 2; t) 2π 1/2 (n + 1)!

(10.54)

t(|n|−1)/2 Γ(|n| − 1/2) F (−1/2, |n| − 1/2; |n|; t) π 1/2 (|n| − 1)!

(10.55)

we have for n ≥ 0 an = and for n ≤ −1 an =

By use of the contiguous relations of hypergeometric functions such as [55, chapter 2], F (a, b; c; t) =

m (1 − c)m t−m m (t − 1)m−k F (a − k, b; c − m; t), (b − c + 1)m k

(10.56)

k=0

where (a)m = Γ(a + m)/Γ(a),

(10.57)

these may be rewritten in terms of the complete elliptic integrals of the first and second kind π/2 dφ π 1/2 K(t ) = (10.58) = F (1/2, 1/2; 1; t) 2 1/2 2 (1 − t sin φ) 0


π/2

dφ(1 − t sin2 φ)1/2 =

E(t1/2 ) = 0

π F (−1/2, 1/2; 1; t). 2

¾

(10.59)

Using the abbreviated notations ˜ = 2 K(t1/2 ) and E ˜ = 2 E(t1/2 ) K π π

(10.60)

we have, for example when T < Tc ˜ σ0,0 σ1,1 = E 1 ˜ 2 + 2(t − 1)2 E ˜K ˜ − (t − 1)2 K ˜ 2} σ0,0 σ2,2 = 2 {(5t − 1)E 3t

(10.61) (10.62)

and for T > Tc ˜ − (1 − t)K} ˜ σ0,0 σ1,1 = t−1/2 {E 1 ˜E ˜ + 3(t − 1)2 K ˜ 2 }. σ0,0 σ2,2 = {(5 − t)E˜ 2 + 8(t − 1)K 3t

(10.63) (10.64)

Furthermore as T → Tc each element an has an expansion an ∼

∞

a± n,k |T

− Tc | + ln |T − Tc | k

k=0

∞

k b± n,k |T − Tc |

(10.65)

k=1

and using (10.65) in (10.45) we find that cancellations take place and that as T → Tc ± the determinants DN have the form for both the row and diagonal correlations: DN =

N

2

|T − Tc |k lnk |T − Tc |

∞

j d± k,j |T − Tc |

(10.66)

j=0

k=0

Form factor expansion For T < Tc the determinant DN may also be expressed in an exponential form [56] DN = (1 − t)1/4 eFN where FN =

∞

(2n)

λ2n FN

(10.67)

(10.68)

n=1

with t given by (10.50), (2n)

FN

=

(−1)n+1 n22n

2n

n dzj zjN −1 −1 Q(z2j−1 )Q(z2j−1 )P (z2j )P (z2j ) (10.69) 1 − z z j j+1 j=1 j=1

where λ = 1/π

(10.70)

¾


the contours of integration are |zj | = 1 − , z2n+1 ≡ z1 and P (z) = 1/Q(z) =

1 − α2 z 1 − α1 z

1/2 .

(10.71)

For T > Tc the corresponding result is ˆ

DN = (1 − t)1/4 XN eFN +1

(10.72)

where t is given by (10.53), the function FˆN is given by (10.68) and (10.69) with P (z) and Q(z) replaced by ˆ = [(1 − α1 z)(1 − α−1 z)]−1/2 Pˆ (z) = 1/Qz 2 and XN =

∞

(2n+1)

λ2n+1 XN

(10.73)

(10.74)

n=0

with (2n+1) XN

1 = (2i)2n+1 ×

n+1

n+1 (zjN +1 dzj ) j=1 n

−1 ) Pˆ (z2j−1 )Pˆ (z2j−1

j=1

1

2n

z1 z2n+1

j=1

1 1 − zj zj+1

ˆ −1 ). ˆ 2j )Q(z Q(z 2j

(10.75)

j=1

The exponentials in (10.67) and (10.72) may be expanded and thus we obtain the form factor expressions for the correlation functions. For T < Tc the form factor expansion is DN = (1 − t)1/4 {1 +

∞

(2n)

λ2n fN

}

(10.76)

n=1

with 1 1 = (n!)2 (2i)2n

(2n) fN

×

1≤j≤n 1≤k≤n

2n

−1 −1 dzj zjN Q(z2j−1 )Q(z2j−1 )P (z2j )P (z2j )

j=1

1

1 − z2j−1 z2k

2

(z2j−1 − z2k−1 )2 (z2j − z2k )2 (10.77)

1≤j Tc the form factor expansion is DN = (1 − t)1/4

∞ n=0

with

(2n+1)

λ2n+1 fN

(10.78)


¾

(2n+1)

fN

1 n!(n + 1)!(2i)2n+1 × 1≤j≤n+1 1≤k≤n

2n+1

(dzj zjN )

j=1

2

1 1 − z2j−1 z2k

n+1

−1 ˆ −1 z2j−1 ) P (z2j−1 )Pˆ (z2j−1

j=1

n

ˆ −1 ) ˆ 2j )Q(z z2j Q(z 2j

j=1

(z2j−1 − z2k−1 )2

1≤j Tc by σN (t) = t(t − 1)

d ln C+ (N, N ) 1 − dt 4

(10.89)

¾


where t is defined by (10.53), then for both T < Tc and T > Tc and all values of λ the function σN satisfies 2 2 d2 σ 1 dσ dσ dσ dσ t(t − 1) 2 −σ −4 −σ− − σ) . (t − 1) t = N 2 (t − 1) dt dt dt dt 4 dt (10.90) We note that, for the Ising case λ = 1/π, C(N, N ; 1/π) must satisfy the normalization condition for T < Tc C− (N, N ; 1/π) = 1 + O(t) for t → 0

(10.91)

and for T > Tc (1/2)N N/2 t (1 + O(t)) for t → 0. N! Equation (10.90) is a special case of the more general equation 2

2 d2 h dh dh dh t(1 − t) 2 2h − (2t − 1) + b1 b2 b3 b4 + dt dt dt dt dh dh dh dh 2 2 2 2 + b1 + b2 + b3 + b4 = dt dt dt dt C+ (N, N ; 1/π) =

(10.92)

(10.93)

with, for example b1 = b4 = N/2, b2 = (1 − N )/2, b3 = (1 + N )/2.

(10.94)

The equation (10.93) has been shown [57, 58] to be equivalent to the Painlevé VI equation 2 d2 q 1 1 1 1 dq dq 1 1 1 + + + + = − dt2 2 q 1−q q−t dt t 1 − t q − t dt t t−1 t(t − 1) q(q − 1)(q − t) α + β (10.95) + γ + δ + t2 (t − 1)2 q2 (q − 1)2 (q − t)2 by the following set of birational transformations [58, (2.5)–(2.7)]:

1 dh q= − b3 b4 C (b3 + b4 )B + 2A dt

where A=

dh + b23 dt

dh + b24 dt

(10.96)

(10.97)

d2 h dh − (b1 b2 b3 + b1 b2 b4 + b1 b3 b4 + b2 b3 b4 ) (10.98) + (b1 + b2 + b3 + b4 ) dt2 dt dh − h − (b1 b2 + b1 b3 + b1 b4 + b2 b3 + b2 b4 + b3 b4 ) (10.99) C=2 t dt

B = t(t − 1) and

where the relation between bj and α, β, γ, δ is given by [58, (1.2)] as b1 + b2 = (−2β)1/2

(10.100)


¾

b1 − b2 = (2γ)1/2 b3 + b4 + 1 = (1 − 2δ)1/2

(10.101) (10.102)

b3 − b4 = (2α)1/2 .

(10.103)

Conversely h is given in terms of q by (2.1) of [58] h = q(q − 1)(q − t)p2 − {b1 (2q − 1)(q − t) − b2 (q − t) + (b3 + b4 )q(q − 1)}p 1 + (b1 + b2 )(b1 + b2 )q − b21 t − (b1 b2 + b1 b3 + b1 b4 + b2 b3 + b2 b4 + b3 b4 ) 2 (10.104) where p is determined in terms of q from (0.6) of [58] ∂h dq = = 2q(q − 1)(q − t)p − {b1 (2q − 1)(q − t) − b2 (q − t) + (b3 + b4 )q(q − 1)}. dt ∂p (10.105) The equation for h is invariant under permutations of bj and the change of any two signs of bj but the transformation equations clearly do not have this symmetry. Thus there are several Painlevé VI equations (10.95) that lead to the same equation for σ. t(t − 1)

Asymptotic behavior as N → ∞ There are three distinct behaviors of C(N, N ) and C(0, N ) as N → ∞: T > Tc , T = Tc and T < Tc which need to be treated separately. The simplest case to consider is C(N, N ) at T = Tc . In this case the determinental representation (10.45) reduces to an N × N Cauchy determinant which reduces to the remarkably simple form N N −1 2 1 σ0,0 σN,N = [1 − 2 ]l−N (10.106) π 4l l=1

and for N → ∞ this behaves as σ0,0 σN,N ∼ AN

−1/4

1−

1 −4 + O(N ) 64N 2

(10.107)

where the transcendental constant A is expressed in terms of the zeta function ζ(z) as A = 21/12 exp[3ζ (−1)] ∼ 0.6450 · · · .

(10.108)

and we note that (10.107) is independent of the ratio E v /E h . For correlations in the same row (column) a more refined computation [13] gives 1/4

1 2 cosh 2Kch −4 σ0,0 σ0,N = A 1+ − 1 + O(N ) N 64N 2 sinh2 (2K v ) 1/4

1 2 cosh 2Kcv −4 1+ − 1 + O(M ) . σ0,0 σM,0 = A M 64M 2 sinh2 (2K h ) (10.109) From both (10.107) and (10.109) we see that the anomalous dimension is η = 1/4.

¾


For both T < Tc and T > Tc the large N behavior is given by the leading terms of the form factor expansions (10.76) and (10.78). Thus on the diagonal we find: for T < Tc (2)

σ0,0 σN,N ∼ (1 − t)1/4 {1 + fN + · · ·} = (1 − t)1/4 {1 +

2tN +1 + · · ·} (10.110) 2πN 2 (t − 1)2

and for T > Tc (1)

σ0,0 σN,N ∼ (1 − t)1/4 fN =

tN/2 1 + ··· 1/2 (πN ) (1 − t)1/4

(10.111)

For correlations in the same row (column) we have for T < Tc (α2 < 1) (2)

σ0,0 σ0,N ∼ (1 − t)1/4 {1 + fN + · · ·} = (1 − t)1/4 {1 +

α2N 1 2 + · · ·} 2πN 2 (α−1 − α2 )2 2 (10.112)

and for T > Tc (α2 > 1) (1)

σ0,0 σ0,N ∼ (1 − t)1/4 fN =

α−N (1 − t)1/4 2 + ··· 1/2 (πN )1/2 (α2 − α−1 2 )

(10.113)

These correlation functions for T = Tc approach their values at N, M → ∞ exponentially rapidly and thus we may define the correlation length on the diagonal ξd by √ tN = e− 2N/ξd± (10.114) and the correlation length in a row ξh by α±N = e−N/ξh± . 2

(10.115)

The correlation in a column ξv is obtained from ξh by interchanging E v ↔ E h . As T → Tc we find from (10.43) and (10.114) that −1 ξd± = 2−1/2 | ln t| ∼ 2−1/2 |1 − k± | √ ∼ 2|β − βc |(E v coth 2E v /kB Tc + E h coth 2E h /kB Tc )

(10.116)

and from (10.30) and (10.115) that −1 ξh± = | ln α2 | ∼ |1 − α2 |

∼ |β − βc |2 tanh 2E v /kB Tc (E v coth 2E v /kB Tc + E h coth 2E h /kB Tc )(10.117) and −1 ξv± ∼ |β − βc |2 tanh 2E h /kB Tc (E v coth 2E v /kB Tc + E h coth 2E h /kB Tc ). (10.118)

These correlation lengths diverge linearly as T → Tc and thus the critical exponent ν = 1.


¾

We see that in general lim ξh± /ξd± = 2−1/2 coth 2E v /kB Tc

T →Tc

lim ξv± /ξd± = 2−1/2 coth 2E h /kB Tc

T →Tc

and that, for E v = E h = E where coth 2E/kB Tc =

√

2,

lim |β − βc |ξh = lim |β − βc |ξv = lim |β − βc |ξd

T →Tc

T →Tc

(10.119)

T →Tc

(10.120)

which is consistent with the expected rotational invariance of the correlation functions of the isotropic lattice near Tc . 10.1.7

The correlation C(M, N ) for general M, N

For general values of M and N it is more efficient to express the correlations as infinite (Fredholm) determinants and from these expressions the following results are obtained [22]: For T < Tc the generalization of σ0,0 σ0,N and σ0,0 σN,N given by (10.67)–(10.71) is σ0,0 σM,N = (1 − t)1/4 exp (−FM,N ) (10.121) where FM,N =

∞

(2n)

FM,N

(10.122)

n=1

and FM,N = (−1)n [2zv (1 − zh2 )]2n (2n)−1 (2π)−4n π π 2n e−iMφj−1 −iN φ2j sin 12 (φ2j−1 − φ2j+1 ) (10.123) dφ1 · · · dφ4n ∆(φ2j−1 , φ2j ) sin 12 (φ2j + φ2j+2 ) −π −π j=1 (2n)

with φ4n+1 = φ1 , φ4n+2 = φ2 , Imφj < 0 and ∆(φ2j−1 , φ2j ) = (1 + zh2 )(1 + zv2 ) − 2zv (1 − zh2 ) cos φ2j−1 − 2zh(1 − zv2) cos φ2j (10.124) and zv = tanh E v /kB T, zh = tanh E h /kB T.

(10.125)

For T > Tc the generalization of σ0,0 σ0,N and σ0,0 σN,N given by (10.72)–(10.75) is σ0,0 σM,N = (1 − t)1/4 XM,N exp (−FM,N ) where XM,N =

∞ n=0

with

(2n+1)

XM,N

(10.126)

(10.127)

¾


(2n+1)

XM,N

= (2π)−4n−2 [2izv (1 − zh2 )]2n

e−iφ4n+1 −iφ4n+2

2n+1 j=1

π

−π

dφ1 · · ·

π

−π

dφ4n+2

2n e−i(M−1)φ2j−1 −i(N −1)φ2j e−iφ2j−1 − e−iφ2j+1 . ∆(φ2j−1 , φ2j ) 1 − eiφ2j +iφ2j+2 j=1

(10.128) As with the row and diagonal correlations these exponentials can be expanded to obtain form factor expansions [22, 36–38] similar to (10.76) and (10.78). Nonlinear difference equations The spin correlations on the lattice also satisfy nonlinear partial difference equations with respect to the locations of the spins and (systems of) differential equations in the temperature. To give these equations we define new constants for what is called the dual lattice by K v∗ and K h∗ by sinh 2K v∗ sinh 2K h = 1 and sinh 2K h∗ sinh 2K v = 1

(10.129)

and let σ0,0 σM,N ∗ be the correlation functions with K v → K h∗ and K h → K v∗ . Then we have [33] σ0,0 σM,N ∗ = (sinh 2K v sinh 2K h )1/2 XM,N σ0,0 σM,N

(10.130)

where XM,N given by (10.127) satisfies the partial difference equation for all M, N except M = N = 0 [∇2L − 2(a − γ1 − γ2 )]XM,N XM+1,N + XM−1,N =− 1 − XM+1,N XM−1,N +1 XM,N +1 XM,N −1 2 × {γ2 [XM,N − XM,N +1 XM,N −1 ] 2 − γ1 XM,N +1 XM,N −1 [XM,N − XM+1,N XM−1,N ]}

+ M ↔ N, γ1 ↔ γ2

(10.131)

where the lattice Laplacian is defined as ∇L XM,N = γ1 [XM+1,N + XM−1,N − 2XM,N ] + γ2 [XM,N +1 + XM,N −1 − 2XM,N ].

(10.132)

The correlations are given in terms of XM,N for T < Tc by (10.121) and for T > Tc by (10.126) where ∞ ln fM,k (10.133) FM,N = k=1

with


fM,N =

∞ 1 − XM+j,N XM+1+j,N +1 j=0

1 − XM+1+j XM+j,N +1

.

¾

(10.134)

The correlations themselves satisfy the two nonlinear partial difference equations σ0,0 σM,N 2 − σ0,0 σM−1,N σ0,0 σM+1,N ' ( = − sinh2 2K v σ0,0 σM,N ∗2 − σ0,0 σM,N −1 ∗ σ0,0 σM,N +1 ∗ (10.135) and σ0,0 σM,N 2 − σ0,0 σM,N −1 σ0,0 σM,N +1 ' ( = − sinh2 2K h σ0,0 σM,N ∗2 − σ0,0 σM−1,N ∗ σ0,0 σM+1,N ∗ . (10.136) At T = Tc we have σ0,0 σM,N = σ0,0 σM,N ∗ and these two equations reduce to the remarkably simple result valid except at M = N = 0 ' ( sinh2 2K v σ0,0 σM,N 2 − σ0,0 σM−1,N σ0,0 σM+1,N ' ( = sinh2 2K h σ0,0 σM,N ∗2 − σ0,0 σM,N −1 ∗ σ0,0 σM,N +1 ∗ . (10.137) Asymptotic behavior for M, N → ∞ For T = Tc there are two cases. Asymptotically as M and/or N → ∞ we have for T < Tc 2 σ0,0 σN,M ∼ M− {1 +

exp(−2M θ1 − 2N θ2 ) + · · ·} 8π(M sinh θ1 cosh θ2 + N cosh θ1 sinh θ2 )2

(10.138)

and for T > Tc 2 σ0,0 σM,N ∼ M+

exp(−M θ1 − N θ2 ) + ··· [2π(M sinh θ1 cosh θ2 + N cosh θ1 sinh θ2 )]1/2

(10.139)

where θ1 is defined by cosh θ1 =

(M 2 /γ1 )(a2 − γ22 ) + N 2 γ1 aM 2 + [M 2 N 2 a2 + (M 2 − N 2 )(M 2 γ22 − N 2 γ12 )]1/2

(10.140)

with a = (1 + zh2 )(1 + zv2 ), γ1 = 2zv (1 − zh2 ) γ2 = 2zh (1 − zv2 )

(10.141)

and θ2 is obtained from θ1 by the interchange M ↔ N, E ↔ E . v

10.1.8

h

Scaling limit

The asymptotic behavior for N, M 1 of the correlation functions derived in the previous subsection is not uniform and the asymptotic expansion at T = Tc cannot be obtained by letting T → Tc in the expansions for T = Tc . In order to smoothly connect

¾


these three different behaviors together we must study the scaling limit introduced in chapter 5. The concept of the scaling limit has been presented in detail in chapter 5. To apply this definition to the Ising model, define scaled coordinates n and m by n = N/ξh , with r=

m = M/ξv

(10.142)

) n 2 + m2 .

(10.143)

The scaling limit is then defined as the limit T → Tc , N → ∞, M → ∞ with n and m, fixed. In this limit the correlation functions are constant on the ellipses of fixed r which, using (10.31), is equivalent to

sinh 2Kch sinh 2Kcv

1/2

M2 +

sinh 2Kcv sinh 2Kch

1/2 N 2 = const.

(10.144)

The scaled two point-function is thus defined as −2 σ0,0 σM,N G± (r) = lim M± scaling

(10.145)

1/8 where M∓ = (1−α±2 . This function is a function of r alone and thus it is sufficient 2 ) to consider the scaling function on the diagonal −2 G± (r) = lim M± σ0,0 σN,N scaling

(10.146)

where σ0,0 σN,N is given by the various expansions of section 10.1.6 with α1 = 0. The scaling limit is obtained by letting α2 → 1 and N → ∞ where for T < Tc we define

and for T > Tc

r = N (1 − α22 )/2 = N (1 − t)/2

(10.147)

r = N (1 − α−2 2 )/2 = N (1 − t)/2

(10.148)

where on the right-hand side we have used the definitions of t of (10.50) and (10.53). Scaled form factor expansion For T < Tc the exponential form of the scaling function is obtained by first deforming the contour in (10.69) from |zj | = 1 to the branch cut which runs from 0 to α2 , and setting zj = α2 xj to obtain 1/2 n (1 − α22 x2j )(x−1 dxj xN 2j − 1) j . 2 −1 2 0 j=1 1 − α2 xj xj+1 j=1 (1 − α2 x2j−1 )(x2j−1 − 1) (10.149) The scaling limit is then obtained by setting 2n(N +1)

(2n) FN

(−1)n+1 α2 = n

2n 1

xj = 1 − yj (1 − α22 ) and using the following approximations which are valid in the scaling limit

(10.150)


2n(N +1)

α2

N xN k = e

= en(N +1) ln α2 → e−n(N +1)(1−α2 ) → e−2nr 2

ln[1−yk (1−α22 )]

2

→e

−N (1−α22 )yk

= e−2ryk

1 − 1 → (1 − α22 )yk x−1 k −1 = 1 − yk (1 − α22 ) 1 − α22 xk = 1 − α22 [1 − yk (1 − α22 )] → (1 − α22 )(1 + yk ) 1 − α22 xj xk = 1 − α22 [1 − (1 − α22 )yj ][1 − (1 − α22 )jk ] → (1 − α22 )[1 + yj + yk ].

¾

(10.151) (10.152) (10.153) (10.154) (10.155)

We thus find from (10.67)–(10.69) and (10.146) that G− (r) = exp

∞

λ2n F˜ (2n) (r)

(10.156)

n=1

where (−1)n+1 e−2nr F˜ (2n) (r) = n

2n ∞

0

j=1

1/2 n dyk e−2ryj y2j (1 + y2j ) (10.157) 1 + yj + yj+1 j=1 y2j−1 (1 + y2j−1 )

with y2n+1 ≡ y1 and λ = 1/π

(10.158)

which is displayed in a more symmetric form by setting yk = (˜ yk − 1)/2

(10.159)

to find (−1)n+1 F˜2n (r) = n

1

1/2 n 2 −1 dyj e−ryj y2j 2 y + yj+1 j=1 y2j−1 −1 j=1 j

2n ∞

(10.160)

(where the tilde on the y has been suppressed). In a similar fashion we find from (10.76) and (10.77) the scaled form factor expansion for T < Tc ∞ G− (r) = 1 + λ2n f˜(2n) (r) (10.161) n=1

with

¿¼¼


f˜(2n) (r) = ×

1 (n!)2

1≤j≤n 1≤k≤n

2n ∞

1

dyj e

n

−ryj

j=1

dyj

j=1

1 (y2j−1 + y2k )2

2 y2j −1 2 y2j−1 − 1

1/2

(y2j−1 − y2k−1 )2 (y2j − y2k )2 (10.162)

1≤j Tc we find from (10.72)–(10.75) ˜ G+ (r) = X(r)G − (r)

(10.163)

where from (10.127) and (10.128)

˜ X(r) =

˜ 2n+1 (r) λ2n+1 X

(10.164)

n=0

with

∞ 2n+1

˜ 2n+1 (r) = (−1)n X 1

j=1

2n n dyj e−ryj 1 (y 2 − 1) (yj2 − 1)1/2 j=1 yj + yj+1 j=1 2j

(10.165)

where again λ = 1/π. From (10.78) and (10.79) we obtain for T > Tc the scaled form factor expansion G+ (r) =

∞

λ2n+1 f˜(2n+1) (r)

(10.166)

n=0

with f

(2n+1)

×

1 (r) = n!(n + 1)!

1≤j≤n+1 1≤k≤n

1

∞ 2n+1 j=1

1 (y2j−1 + y2k )2

dyj e−ryj

n

2 (y2j − 1)1/2

j=1

n+1

2 (y2j−1 − 1)−1/2

j=1

(y2j−1 − y2k−1 )2

1≤j Tc ).


¿¼¿

By expanding the exponential in the expansions for the correlation functions on the lattice (10.121) and (10.126), we can write explicit expressions for (10.183) in terms of integrals kB T χ+ (T ) = (1 − t)1/4 t−1/4

∞

χ ˆ(2j+1) (T ) for T > Tc

(10.184)

j=0

kB T χ− (T ) = (1 − t)1/4

∞

χ ˆ(2j) (T ) for T < Tc

(10.185)

j=1

where (j)

χ ˆ

cotj α (T ) = j!

with

π

−π

dω1 ··· 2π

π

−π

dωj−1 2π

j

1 sinh γn n=1

H (j)

1+ 1−

j n=1 j

) xn = cot2 α ξ − cos ωn − (ξ − cos ωn )2 − (cot α)−4 ) sinh γn = cot2 α (ξ − cos ωn )2 − (cot α)−4

where cot α =

) sh /sv

xn

(10.186)

(10.187) (10.188) (10.189)

1/2 (1 + s2v )1/2 ξ = (1 + s−2 h )

sv = sinh 2E v /kB T

n=1

xn

sh = sinh 2E h /kB T

(10.190) (10.191)

and ωj is defined in terms of the remaining ωi from ω1 + · · · ωj = 0 mod 2π. There are many equivalent expressions H (j) . The original expression [22] comes directly from the expansion of the exponential. Subsequent developments [63–66, 36] have discovered more compact explicit expressions. Reference [63] is for the isotropic lattice E v = E h . Refeferences [64,65] extend the results to the anisotropic case and [38] writes the integrals of [64–66] in terms of trigonometric instead of elliptic functions. We list here the expression of [38]:  H (j) = 

2 hik 

(10.192)

1≤i Tc (T < Tc ). Then if we further use the fact that the corrections to the T = Tc correlation function are of order O(R−9/4 ) (and not O(R−5/4 )) we find that [21, 22] χ ˆ(2) (T ) =

kB T χ(T ) ∼ C0± |1 − Tc /T |−7/4 + C1± |1 − Tc /T |−3/4 + D + o(1) where

C1− C1+ =− = −R0 C0+ C0−

(10.199) (10.200)

with R0 =

2 2 4 2 2 4 (1 + 6zhc + zhc ) + Ev2 zhc (1 + 6zvc + zvc ) − 8Ev Eh zvc zhc (zvc + zhc )2 Eh2 zvc , 8zvc zhc (zvc + zhc )kB Tc [Eh (1 − zhc ) + Ev (1 − z vc )] (10.201)

and C0± = 2−1/2 coth 2Kcv coth 2Kch [Kcv coth 2Kcv + Kch coth 2Kch ]−7/4 I±

0 } 1 0 and the integrals have been numerically evaluated to 52 digits in [38]: where

I± = π2−1/2

∞

dr r{G± (r) −

(10.202) (10.203)

I+ = 1.000815260440212647119476363047210236937534925597789 (10.204) 1.000960328725262189480934955172097320572505951770117 (10.205) I+ = 12π To compute the constant D in (10.199) short distance contributions not included in the scaling functions G± (r) must be taken into account. This was done numerically in [67] where it was found that D = −0.05365771128w−1 + 0.006362291w − 0.00000132w3 with w =

tanh 2Kcv

tanh 2Kch

(10.206)

and all coefficients are accurate to nine significant figures.


¿

Finally we note that all correlations σ0,0 σM,N have logarithmic singularities at T = Tc which must be present in the susceptibility and that there will in addition be an infinite number of “corrections to scaling”. These two effects have been studied in detail in [38] where it is shown that, for the isotropic lattice in terms of the temperature variable τ = (s−1 − s)/2, √ kB T χ± (t) = I± (2Kc 2)−7/4 |τ |−7/4 F± (τ ) + B (10.207) where F± (τ ) = 1 + (j)

5τ 2 3τ 3 23τ 4 35τ 5 (j) j τ + + − − + f± τ 2 8 16 384 768 j=6

(j)

(14)

where f+ = f− and terms as far as f+

(15)

and f−

(10.208)

have been computed and

√

B=

q] ∞ [

bp,q τ q (ln |τ |)p

(10.209)

q=0 p=0

where the terms in (10.209) have been numerically evaluated up to order ln3 τ. But what is most interesting about the susceptibility χ(T ) and makes it much more complicated than either the free energy or the spontaneous magnetization is that in addition to the singularity at Tc there are many other singularities in χ ˆ(j) (T ) in the complex T plane. These new singularities were first discovered in the isotropic case E v = E h by Nickel [36] in 1999 and found in the anisotropic case by Orrick, Nickel, Guttmann and Perk [38] in 2001. These singularities occur because the integrals χˆj (T ) of (10.186) will be singular at the symmetry points of the integrand and where the denominator factor 1 − jn=1 xn vanishes. The symmetry condition requires all ωn to be equal and given by ωn = ω = 2πm /j with m = 1, 2, · · · , j. The vanishing of the denominator factor requires the xn , now all equal, to be given by xn = x = exp(2πim/j) with m = 1, 2, · · · j. From the explicit formula for xn (10.187) we find cot2 α(ξ − cos(2πm /j)) = cos(2πm/j)

(10.210)

or substituting (10.189) for α and (10.190) for ξ we find that there are singularities in χ ˆ(j) at (complex) temperatures where cosh 2E v /kB T cosh 2E h /kB T − sinh 2E h /kB T cos(2πm /j) − sinh 2E v /kB T cos(2πm/j) = 0

(10.211)

which remarkably is exactly the condition (10.9) for the location of the zeros of the (j) partition function. If we call the deviation from the singular temperatures Tm,m determined by (10.211) then for T > Tc the singularity in χ ˆ(2j+1) (T ) is shown in [36] to be 2j(j+1)−1 ln (10.212) ˆ(2j) (T ) shown in [37] to be and for T < Tc the singularity in χ

¿


2j

2

−3/2

.

(10.213)

These singularities become dense as j → ∞ and consequently if there are no cancellations there will be a natural boundary in χ(T ) at the location of the zeros of the partition function. For arbitrary values of the parameter λ in the form factor expansions such as (10.76) and (10.78) where each n particle contribution is weighted by λn such cancellations are generically impossible. But for the Ising model the value of λ is not arbitrary, and it is surely possible that for this value cancellations can occur. The study of this problem is most important because the existence of a natural boundary would be a new phenomenon not previously seen. We conclude this section by noting that in [38] it is proven that χˆ(j) must satisfy a linear differential equation in the temperature. Such a function is called D-finite. The theorem in [38] is an existence proof, and in that paper no examples beyond the elementary cases of χ(1) (T ) and χ(2) (T ) were known. However, in 2004 it was found [41] that in terms of the variable 1 w= (10.214) 2(s + s−1 ) χ ˆ(3) (w) satisfies the seventh-order equation 7

Pn (w)

n=0

dn (3) χ ˆ (w) = 0 dwn

(10.215)

where P0 (w), · · · , P7 (w) are polynomials in w of degrees 36, 41, 42, 43, 44, 45, 46 and 47 respectively. All the singularities in (10.215) are regular and thus the differential equation is Fuchsian. For χ(4) (T ) a Fuchsian equation of tenth order has been also found [43]. The equation (10.215) was discovered by generating the first 490 terms in the power series expansion of χ ˆ(3) (w) in terms of w. The equation is determined by the first 359 of these coefficients and thus there are 131 verifications that (10.215) is correct. The challenge now is to find an analytic way to produce differential equations for all χ ˆ(j) (w) and to see if it is possible to find some equation (differential, difference or functional) that can characterize the full susceptibility in the way in which the Painlevé VI equation characterizes the diagonal two-point function. 10.1.10

The diagonal susceptibility

The expression (10.186) for χ ˆ(j) (T ) is cumbersome and to gain further insight it is useful to consider the simpler problem where the magnetic field interacts with the spins of only one diagonal of the lattice [45]. The susceptibility of a spin on the diagonal in response to this diagonal magnetic field is expressed in terms of the diagonal spin correlations in analogy with (10.183) as kB T χd (T ) =

∞

2 {C(N, N ) − M− (T )}.

(10.216)

N =−∞

Thus if we use the form factor expansions (10.76)–(10.79) we find in analogy to (10.184)–(10.186) that for T > Tc


kB T χd+ (T ) = (1 − t)1/4

∞

(2n+1)

χ ˆd

(t)

¿

(10.217)

n=0

where (2n+1) (t) χ ˆd

tn(n+1) = n!(n + 1)!π 2n+1

1 2n+1

0

k=1

n+1

−1/2 [x2j−1 (1 − tx2j−1 )(1 − x−1 2j−1 )]

j=1

1≤j≤n+1 1≤k≤n

1 − tx2j−1 x2k

n

1−

tn+1/2

2n+1 k=1 2n+1 k=1

xk xk

[x2j (1 − tx2j )(1 − x2j )]1/2

j=1

2

1

dxk

1 + tn+1/2

(x2j−1 − x2k−1 )2

1≤j zv (1 − α2 )/(1 + α2 ) we find σ1,0 σ1,N ∼ M12 (Hb ) + Ab−2 α2N /N 5

(10.267)

where the amplitudes Ab+ , Ab−1 , Ab−2 are rather tedious functions of E h /kB T, E v /kB T and Hb /kB T. For T = Tc if Hb = 0 we have σ1,0 σ1,N ∼

1 πzvc N

(10.268)

and if Hb = 0 σ1,0 σ1,N ∼ M12 (Hb ) +

2 4z 2 zvc . 2 − z 2 )2 (1 − r)2 (1 − r−1 )2 N 4 π 2 (zvc

(10.269)

These large N expansions are not uniform, and to connect the various regions together appropriate scaling variables and functions must be used. To connect the

¿


expansion of T > Tc and T < Tc with the expansions for T − Tc the scaling variable (1 − α2 )N is used, and the scaling functions are expressed in terms of Bessel functions. To connect the T = Tc expansions for Hb = 0 with Hb = 0 the scaling variable N z 2 is used, and the scaling functions are given in terms of functions such as the cosine integral ∞ cos t . (10.270) Ci(z) = − dt t z 10.2.4

Analytic continuation and hysteresis

The boundary magnetization M1 (Hb ) has one further property that deserves to be discussed. Namely, even though for T < Tc the magnetization is discontinuous in the sense that lim − M1 (Hb ) = − lim + M1 (Hb ), (10.271) Hb →0

Hb →0

the boundary magnetization M1 (Hb ) defined for Hb > 0 is analytic at Hb = 0 and can be analytically continued into the region Hb < 0 to a function M1c (Hb ) which is not the same as M1 (Hb ). This analytic continuation may be made by writing M1 (Hb ) in the form 1 − z2 M1 (Hb ) = z + 2π

π

dθ −π

1 + e−iθ 1 + eiθ + s(eiθ ) s(e−iθ )

(10.272)

where s(eiθ ) = 2z(1 + eiθ ) − (1 + zv ){[(1 − α1 eiθ )(1 − α2 eiθ )]1/2 − eiθ [(1 − α1 e−iθ )(1 − α2 e−iθ )]1/2 }. (10.273) The functions s(e±iθ ) have at most one zero in the cut eiθ plane. For Hb small and positive s(eiθ ) has a zero inside the unit circle at r where |r| < 1 and s(e−iθ ) has a zero outside the unit circle at 1/r. After analytic continuation to small negative values of Hb we have r > 1. For Hb < 0 the difference between M1c (Hb ) and M1 (Hb ) is due to the residues at r and 1/r and we find from (10.273) that M1c (Hb ) − M1 (Hb ) =

2z[(1 + zh )2 (zv2 − z 2 )2 − zv2 (1 − zh )2 (1 − z 2 )2 ] . (r−1 − r)(1 − z 2 )zh (zv2 − z 2 )2

(10.274)

When Hb is small and negative the right-hand side of (10.274) is positive and decreases as Hb becomes more negative. The right-hand side of (10.273) vanishes when Hb reaches a critical value Hbc defined by tanh2 Hbc /kB T = zv

1 − α2 . 1 + α2

(10.275)

We plot M1 (Hb ) and the analytic continuation M1c (Hb ) together in Fig. 10.4. It is natural to interpret this figure as a hysteresis loop. The loop shrinks to a point at

Boundary properties of the homogeneous lattice at H = 0

M

¿

M1

McI Hbc

Hb

Z2 = |Zv|(1–α2)/(1+α2)

Fig. 10.4 Hysteresis loop for the magnetization in the first row for E v = E h at T /Tc = 0.9. The solid curve is M1 ; the dotted curve shows the analytic continuation. +1 J = 100

M¥

J=2 J=1

M –Hbc

Hb Hbc

–1

Fig. 10.5 A schematic plot of Mj (Hb ) and its analytic continuation for E v = E h versus Hb for various j.

Hb = M1 = 0 as T → Tc and as T → 0 the loop becomes a rectangle that continues as far into the unstable regime as |Hb | = Ev . Further insight into this hysteresis phenomenon can be obtained by computing the magnetization Mj (Hb ) on the row j in from the boundary. The results of this computation are plotted in Fig. 10.5 where we give the hysteresis loops for Mj (Hb ) as a function of Hb and in Fig. 10.6 where we plot Mj (Hb ) as a function of j for various Hb . The critical value Hbc at which Mjc (Hbc ) = Mj (Hbc ) is independent of j, and in Fig. 10.6 we see that as Hb → Hbc that the region of overturned spins penetrates into the bulk. When Hb reaches Hbc no further continuation is possible because the bulk

¿


magnetization flips from positive to negative. This phenomenon of the boundary field affecting the bulk spins in the interior is sometimes referred to as “wetting.” 1

Hb = ¥

M¥

H b = 0+

J

M 0

STABLE METASTABLE

Hb = –Hbc –M¥

Hb = – ¥

Fig. 10.6 A schematic plot of Mj (Hb ) and its analytic continuation for E v = E h versus j c for various Hb . Metastable values between Hb = 0 and −HB which are reached from positive values of Hb are shown in dotted lines.

10.3

The layered random lattice

Thus far we have considered only homogeneous models where the interaction energies E v and E h are constant throughout the entire lattice. But real systems will contain impurities and will not have this homogeneous property. Therefore it is very desirable to be able to study what new effects can be present if the assumption of homogeneous interactions is lifted. The effect of inhomogeneities can be studied in the Ising model by letting the interaction constants depend on the position in the lattice. The case which is best studied is the case of the layered Ising model where E v and E h are the same within a row where the values in different rows are allowed to be different. For this layered model the interaction energy is E=−

Lh Lv

{E h (j)σj,k σj,k+1 + E v (j)σj,k σj+1,k + Hσj,k }.

(10.276)

j=1 k=1

The case where the energies E v (j) and E h (j) depend on j but still have a periodicity E v (j + J) = E v (j) and E v (j + J) = E v (j) has been studied in [69]. It is found that the specific heat still has a logarithmic singularity at a well defined Tc which is determined from J h e4E (j)/kB Tc tanh2 E v (j)/kb Tc = 1 (10.277) j=1

The layered random lattice

¿

which is invariant under permutations of the E v (j) and E h (j). The amplitude of the logarithmic singularity depends in detail on the values of E v (j) and E h (j) and is not invariant under permutations. A more interesting case, however, is when the E v (j) and E h (j) are quasiperiodic functions of j. A particularly nice example is the Fibonacci lattice [70] which is defined as follows. Consider a set of sequences Sn of the letters A and B defined recursively by Sn+1 = Sn Sn−1 with S0 = 1, S1 = A.

(10.278)

For example S2 = AB, S3 = ABA and S4 = ABAAB. The sequence Sn contains Fn−1 A s and Fn−2 B s where Fn are the Fibonacci numbers defined by Fn+1 = Fn + Fn−1 , F0 = F1 = 1.

(10.279)

v v Now consider two values of (positive) energies E v = EA and EB , place them in the lattice according to the sequence Sn and then repeat the sequence to build up the entire lattice. Each sequence Sn thus defines a set of interactions with periodic Fn+1 from which Tc is computed from (10.277) as

e−2Fn E

h

/kB Tc

v h = tanhFn−1 EA /kB Tc tanhFn−2 EB /kB Tc

(10.280)

and for each of these lattices the specific heat has a logarithmic divergence. In the limit where the sequence length n → ∞ the critical temperature is found from (10.280) as e−2E where

h

/kB Tc

2

v = tanhα E v /kB Tc tanhα EB /kB Tc

√ α = ( 5 − 1)/2

(10.281)

(10.282)

and the specific heat still has a logarithmic divergence with the amplitude x2 ln x2 1 −1 −1 v v 2 {2E h + EA (z − z ) α(zvAc − zvAc ) + EB α (zvBc − zvBc )}2 hc 4π(kb Tc )2 hc 1 − x2 (10.283) where x = zvBc /zvAc . Thus the mere destruction of periodicity is not sufficient to destroy the logarithmic divergence in the specific heat. The most interesting case of all is when these energies E h (j) and E v (j) are allowed to be chosen in a random fashion with probability distributions Pv (E v ) and Ph (E h ). This is the case of frozen (or quenched) random impurities which should be most relevant for real ferromagnets on impure lattices. These frozen random layered Ising models have been extensively investigated [16– 18], and many interesting properties have been found. There is now a new length scale in the problem determined by the width of the probability distributions. As an example we consider the case where E h is fixed and A=

¿


the distribution function for E v is P (E v ). Then there is still a Tc which is determined from (10.277) in the random limit as ∞ h 0= dE h P (E v ) ln(e4E /kB Tc tanh2 E v /kB Tc ). (10.284) 0

Near this Tc we define a temperature variable δ which is useful for sufficiently narrow P (E v ) ∞ h 2 0 dE v P (E v ) ln(e4E /kB T tanh2 E v /kB T ) δ = ∞ (10.285) v v 4E h /kB T tanh2 E v /k T )]2 B 0 dE P (E )[ln(e and we find that for narrow P (ev ) that when δ is of order one that the leading contribution to the specific heat which depends on δ is ∞ ∂2 dφ[ 2 ln Kδ (φ) − (1 + φ)−1 ]. (10.286) ∂δ 0 This function is analytic except at δ = 0 where there is an essential singularity because, while all derivatives exist at δ = 0, the Taylor series fails to converge. Further insight into the behavior of this random system is provided by a general theorem proven specifically by Griffiths [19] for a lattice where all interactions (not just in layers) are random with the specific probability distribution P (E) = pδ(E − E0 ) + (1 − p)δ(E).

(10.287)

Griffiths proved, that for all temperatures below the temperature where the lattice would be critical for the pure case p = 0, the magnetization M (H) will not be analytic at H = 0 even if T > Tc . There is nothing in this theorem which is limited to the specific distribution (10.287) or which cannot be extended to the layered case. The essence of the physics is that if the probability distribution P (E) is nonzero only for E L < E < E U then, if TL < T < TU

(10.288)

where T L (T U ) is the critical temperature the lattice would have if all interactions E take on the single value E U (E L ), there will be large regions of the random lattice which are locally above (below) the actual global value of Tc . It is in the temperature range (10.288) that the computation of the specific heat (10.286) is valid and Griffiths theorem shows that M (H) is not analytic at H = 0. These two results strongly suggest that the zeros of the finite size partition function are no longer pinching the real temperature axis at a single point Tc but are instead pinching the entire line segment (10.288). What is not shown by these computations is 1) the connection of Tc defined by (10.284) with the temperature below which there is spontaneous magnetization and 2) the nature of the nonanalyticity in M (H). The first question has been studied [71] for the layered random lattice by means of renormalization group techniques where he finds that the temperature for the onset

The Ising model for H = 0

¿

of spontaneous magnetization is indeed the Tc computed √ from (10.284), but that the exponent β for the random layered lattice is now (3 − 5)/2. The question of the nature of the nonanalyticity has only been studied for the average magnetization on the boundary [17,18] where it is found that when the temperature variable δ is of order one that there are terms in the average boundary magnetization M1 (Hb ) which depend on H 2|δ| which are not analytic (unless 2|δ| is an integer) and which even fail to be differentiable for |δ| < 1/2. Thus there is an entire region around Tc where the boundary magnetic susceptibility is infinite. Furthermore in the temperature region (10.288) the average over P (E v ) of the boundary spin correlations falls off at large separations as a power of the separation of the spins which depends on δ. We thus conclude that in the temperature region (10.288) the standard phenomenological description of critical phenomena does not apply.

10.4


The desire to extend the exact computation of the two-dimensional Ising model from H = 0 to H = 0 has been one of the outstanding problems in statistical mechanics ever since Onsager’s computation of 1944. 10.4.1

The circle theorem

The first result obtained for H = 0 is the famous circle theorem of Lee and Yang [46] which states that in terms of the variable z = e−2H/kB T the zeroes of the partition function of an Ising model with all interaction energies E positive (i.e. ferromagnetic) and with boundary conditions which are invariant under the reflection H → −H for real temperatures on a finite size lattice in D dimensions lie on the unit circle |z| = 1. In other words, in the magnetic field plane the partition function is periodic with a period iπkB T and the zeros of the partition function all lie exactly on the imaginary H axis. When T > Tc there is a strip surrounding the real H axis which is zero-free but for T < Tc the zeros pinch the real H axis at H = 0 and the density of these zeros of the spontaneous magnetization. 10.4.2

The imaginary magnetic field H/kB T = iπ/2

Lee and Yang also found [46] that for H/kB T = iπ/2

(10.289)

the free energy of the Ising model can also be exactly solved for E v = E h . This solution was extended to the anisotropic case in 1966 where [47] it was found that free energy is π π 1 v h −F (ikB T π/2)/kB T = ln(2 cosh E /kB T cosh E /kB T ) + dφ1 dφ2 16π 2 −π −π × ln[4(zv2 + zh2 )(1 + zv2 zh2 ) − 4zv2 (zh2 − 1)2 cos2 φ1 − 4zh2 (zv2 − 1)2 cos2 φ2 ] with zj defined by (10.125), and that the magnetization is

(10.290)

¿¾¼


(zv−1 + zv )2 (zh−1 + zh )2 M (ikB T π/2) = 4(zv−2 + zv2 + zh−2 + zh2 )

1/8 .

(10.291)

The free energy (10.290) is analytic for all real T > 0. The magnetization (10.291) is the density of zeros at H/kB T = iπ/2 (y = −1) and for positive T is a monotonic function with the limiting values M (ikB T π/2) → 1 as T → ∞

(10.292)

→ ∞ as T → 0.

(10.293)

The results for the free energy extend to the finite size partition function with toroidal boundary conditions as the sum of four terms. For the isotropic lattice we find in analogy with the H = 0 result that the partition function normalized to unity at u = x2 = e−4K = 0 is in the form (10.2) with Zj replaced by Zj (u, z = −1) = (1 − u)Lv Lh /2 {1 + u2 + u[6 − 4 cos2 θ1 − 4 cos2 θ2 ]}. (10.294) θ1

θ2

For Brascamp–Kunz boundary conditions we have [72] a result analogous to (10.19) for the partition function normalized to unity at u = 0 when both Lv and Lh are even:

Lh /2 Lv /2

ZBK (u, z = −1) = (1 − u)Lv Lh /2

{1 + u2

j=1 k=1

+ u[6 − 4 cos ((2j − 1)π/Lh ) − 4 cos2 ((2k − 1)π/(2(Lv + 1)))]} (10.295) 2

If Lh /4 is an integer this further reduces to a perfect square

Lh /4 Lv /2

ZBK (u, z = −1) = (1 − u)

Lv Lh /2

{1 + u2

j=1 k=1

+ u[6 − 4 cos2 ((2j − 1)π/Lh ) − 4 cos2 ((2k − 1)π/(2(Lv + 1)))]}2 .(10.296) We note that, because of the factor (1 − u)Lv Lh /2 , the partition functions for both toroidal and Brascamp–Kunz boundary conditions have half of the zeros at the point u = 1. We also note that for the Brascamp–Kunz boundary conditions all the remaining zeros lie either in the circle |u| = 1 where − 1 ≤ 3 − 2 cos2 θ1 − 2 cos2 θ2 ≤ 1 or on the segment of the negative real axis √ √ −3 − 2 2 ≤ u ≤ −3 + 2 2 where 1 ≤ 3 − 2 cos2 θ1 − 2 cos2 θ2 .

(10.297)

(10.298)

A plot of the zeros of a 12 × 12 lattice with Brascamp-Kunz boundary conditions is given in Fig. 10.7. Partition function zeros for finite size lattices at H/kB T = iπ/2 with other boundary conditions have been studied in [54].


¿¾½

1.0

−5

−4

−3

−2

−1

1 −1.0

Fig. 10.7 Zeros in the complex u = x2 plane of the Ising partition function at H/kB T = iπ/2 for Brascamp-Kunz boundary conditions on a 12 × 12 isotropic lattice. The zero at x = 1 has multiplicity 72 and all other zeros have multiplicity 2.

10.4.3

Expansions for small H

For H = 0 there are no further results known, and to obtain further information we turn to expansions of quantities such as the free energy and correlation functions as power series in H. Free energy. The free energy F (T ; H) is expanded as n+1 ∞ H −[F (T ; H) − F (T ; 0)]/kB T = σ0,0 σR1 · · · σRn c k T B n=0

(10.299)

R1 ,···,Rn

where σ0,0 σR1 · · · σRn c denotes the connected part of the correlation function which vanishes when the separation of any two spins goes to infinity. The first two terms in this expansion are the magnetization and the susceptibility which have been previously discussed. For T > Tc all terms in (10.299) with n even vanish. Furthermore the circle theorem of Lee and Yang [46] guarantees that F (T, H) is analytic at H = 0 and thus the series (10.299) will converge. For T < Tc it was shown by Isakov [49] in 1984 that in dimension two and greater that the derivatives ∂ k F (T, H)/∂H k increase with k sufficiently rapidly that the series in (10.299) will not converge. Therefore for T < Tc the free energy and magnetization are not analytic at H = 0 and unlike the boundary magnetization as a function of the boundary magnetic field the magnetization M (T ; H) cannot be analytically continued from H > 0 into a metastable (hysteresis) phase for H < 0. Two-point function. The two-point function in a magnetic field G(R1 − R2 ; H) is expanded in terms of the n point correlations at H = 0 as G(R1 − R2 ; H) − M (T, H) = 2

∞

(H/kB T )n

n=0

σR1 σR2 σR3 · · · σRn+2 c

R3 ···Rn+2

(10.300) and we may use the extensive information available [28–30] about n point functions at H = 0 to study the region T ∼ Tc . In the scaling limit T → Tc and Rj − Rk → ∞, (10.301)

¿¾¾


assuming with no loss of generality that E v = E h , (Rj − Rk )|T − Tc | = rj − rk fixed

(10.302)

it was shown in [29, 30] that at H = 0 the correlation functions have the form n σR1 · · · σRn ∼ M± g(r1 , · · · , rn ).

(10.303)

Therefore by using the scaling form (10.303) in (10.300) and recalling that M± (T ) ∼ |T − Tc |1/8 as T → Tc we see that if we define a scaled magnetic field as h = H/|T − Tc |15/8 then

(10.304)

−2 lim M± G(R1 , · · · R2 ; H)c = g c (r; h)

scaling

(10.305)

is a function of r and h alone. The most interesting case is T < Tc where the connected two-point function g c (r; h) for h ∼ 0 and large r was found in [48] to be g c (r; h) = aj (h)K0 [(2 + κj (h))r] ∼ π 1/2 r−1/2 e−2r aj (h)erκj (h) (10.306) j

j

with κj (h) = h2/3 λj and aj (h) = ha

(10.307)

where a is a constant and λj are the solutions of J1/3 (λ3/2 /3) + J−1/3 (λ2/3 /3) = 0

(10.308)

where Jn (z) is the Bessel function of order n. There are an infinite number of solutions to (10.308). As h → 0 the spacing between these zeros vanishes and the large r behavior (10.306) reduces to the behavior r−2 e−2r of the two-point function at H = 0 for T < Tc . If we consider the two-dimensional Fourier transform of g c (r, h) g˜c (k, h) =

d2 reik·r =

2π

∞

dθ 0

dr rekr cos θ g c (r; h)

(10.309)

0

we see from (10.306) that the singularities in g c (k; h) nearest the real k axis are poles. In the language of field theory the positions of these poles may be interpreted as particle masses. 10.4.4

T = Tc with H > 0

We have now exhausted the known results for the Ising model (10.1) on a lattice with H = 0. However, if we are willing to consider only the scaling limit where the lattice has disappeared and continuum methods are used then in a most remarkable paper Zamolodchikov [50] discovered in 1988 that the locations mi of the poles in the k plane of the Fourier transform (10.309) of the two-point function are given by the


¿¾¿

Table 10.5 The location of the poles in the Fourier transform of the two-point function of the scaled Ising model for T = Tc and H > 0.

m1 m2 m3 m4 m5 m6 m7 m8

=m = 2m cos π/5 = 2m cos π/30 = 4m cos π/5 cos 7π/30 = 4m cos π/5 cos 2π/15 = 4m cos π/5 cos π/30 = 8m cos2 π/5 cos 7π/30 = 8m cos2 π/5 cos 2π/15

=m = m1.61803 · · · = m1.98904 · · · = m2.40487 · · · = m2.95629 · · · = m3.21834 · · · = m3.89116 · · · = m4.78338 · · ·

eight components Sk of the Perron–Frobenius vector of the Cartan matrix of the Lie algebra E8 as mj /mk = Si /Sk . We list these poles in Table 10.5. In the language of field theory this spectrum of poles has three particles below the first two-particle threshold and five particles above threshold. Nevertheless the five particles above the two-body threshold are stable and do not decay. However, if (in the field theory limit) the system with H > 0 is perturbed from T = Tc it is found [73] that most of the decays compatible with energy conservation do occur. The exceptional cases that are still forbidden are m7 → m1 m1 and the four decays m8 → m 1 m 1 , m1 m 2 , m2 m 3 , m3 m 4 . 10.4.5

Extended analyticity

We conclude by noting that if it is assumed [51] that the free energy in the scaling limit, which depends on the single variable h and not on the two separate variables H and T , can be analytically continued through the position of the Lee–Yang zeros then a scenario is obtained which joins together all of the regions previously discussed. To fully evaluate this conjecture it is necessary to determine whether or not the singularities found in χ(j) discovered in [36–39] actually lead to a natural boundary.

References [1] W. Lenz, Phys. ZS. 21 (1920) 613. [2] E. Ising, Beitrag zur theorie des ferromagnetismus, Z. Physik 31 (1925) 253–258. [3] R. Peierls, On Ising’s model of ferromagnetism, Proc. Cambridge Phil. Soc. 32 (1936) 477–481, [4] H.A. Kramers and G.H. Wannier, Statistics of the two-dimensional ferromagnet. Part I, Phys. Rev. 60 (1941) 252–262. [5] H.A. Kramers and G.H. Wannier, Statistics of the two-dimensional ferromagnet. Part II, Phys. Rev. 60 (1941) 263–276. [6] L. Onsager. Crystal statistics I. A two dimensional model with an order disorder transition, Phys. Rev. 65 (1944) 117–149. [7] L. Onsager, discussion, Nuovo Cimento 6 Suppl. (1949) 261. [8] B. Kaufman, Crystal statistics II. Partition function evaluated by spinor analysis, Phys. Rev. 76 (1949) 1232–1243. [9] B. Kaufman and L. Onsager, Crystal statistics III short range order in a binary Ising lattice, Phys. Rev. 76 (1949) 1244–1252. [10] C.N. Yang, The spontaneous magnetization of the two dimensional Ising model, Phys. Rev. 85 (1952) 808–816. [11] P.W. Kastelyn, Dimer statistics and phase transitions, J. Math. Phys. 4 (1963) 287–293. [12] E.W. Montroll, R.B. Potts and J.C. Ward, Correlations and spontaneous magnetization of the two dimensional Ising model, J. Math. Phys. 4 (1963) 308–322 [13] T.T. Wu, Theory of Toeplitz determinants and the spin correlations of the two dimensional Ising model I, Phys. Rev. 149 (1966) 380–401. [14] H. Cheng and T.T. Wu, Theory of Toeplitz determinants and the spin correlations of the two dimensional Ising model III, Phys. Rev. 164 (1967) 719–735. [15] B.M. McCoy and T.T. Wu, Theory of Toeplitz determinants and the spin correlations of the two dimensional Ising model IV, Phys. Rev. 162 (1967) 436–475. [16] B.M. McCoy and T.T. Wu, Theory of a two dimensional Ising model with random impurities, Phys. Rev. 176 (1968) 631–643. [17] B.M. McCoy, Incompleteness of the critical exponent description for ferromagnetic systems containing random impurities, Phys. Rev. Letts, 23 (1969) 383–386. [18] B.M. McCoy, Theory of a two dimensional Ising model with random impurities III. Boundary effects, Phys. Rev. 188 (1969) 1014–1031. [19] R.B. Griffiths, Nonanalytic behavior above the critical points in a random Ising ferromagnet, Phys. Rev. Letts. 23 (1969) 17–19. [20] E. Barouch, B.M. McCoy and T.T. Wu, Zero-field susceptibility of the two dimensional Ising model near Tc , Phys. Rev. Letts. 31 (1973) 1409–1411.

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[21] C.A. Tracy and B.M. McCoy, Neutron scattering and the correlations functions of the Ising model near Tc , Phys. Rev. Letters. 31 (1973) 1500–1504. [22] T.T. Wu, B.M. McCoy, C.A. Tracy and E. Barouch, Spin-spin correlation functions for the two dimensional Ising model: exact theory in the scaling region, Phys. Rev. B13 (1976) 315–374. [23] M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields I, Pub. Res. Math. Sci. 14 (1978) 223–267. [24] M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields II, Pub. Res. Math. Sci. 15 (1979) 201–278. [25] M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields III, Pub. Res. Math. Sci. 15 (1979) 577–629. [26] M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields IV, Pub. Res. Math. Sci. 15 (1978) 871–972. [27] M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields V, Pub. Res. Math. Sci. 16 (1980) 531–584. [28] B.M.McCoy, C.A. Tracy and T.T. Wu, Two-dimensional Ising model as an exactly solved relativistic quantum field theory: Explicit formulas for the N − point functions, Phys. Rev. Letts. 38 (1977) 793–796. [29] B.M. McCoy and T.T. Wu, Two-dimensional Ising field theory for T < Tc : string structure of the three point function, Phys. Rev. D 18 (1978) 1243–1252. [30] B.M. McCoy and T.T. Wu, Two-dimensional Ising field theory for T < Tc : Greens function strings in strings in n point functions, Phys. Rev. D 18 (1978) 1253–1258. [31] B.M. McCoy and T.T. Wu, Nonlinear partial difference equations for the twodimensional Ising model, Phys. Rev. Letts. 45 (1980) 675–678. [32] J.H.H. Perk, Quadratic identities for the Ising model, Phys. Letts.A 79 (1980) 3–5. [33] B.M. McCoy and T.T. Wu, Non-linear partial difference equations for the two-spin correlation function of the two dimensional Ising model, Nucl. Phys. B 180[FS2] (1981) 89–115. [34] B.M. McCoy, J.H.H. Perk and T.T. Wu, Ising field theory: quadratic difference equations for the n-point Green’s functions on the lattice, Phys. Rev. Letts. 46 (1981) 757–760. [35] M. Jimbo and T. Miwa, Studies on holonomic quantum fields XVII, Proc. Jpn. Acad. 56A (1980) 405; 57A (1981) 347. [36] B.G. Nickel, On the singularity structure of the susceptibility of the 2D Ising model, J. Phys. A32 (1999) 3889–3906. [37] B.G. Nickel, Addendum to “On the singularity structure of the susceptibility of the 2D Ising model,” J. Phys. A 33 (2000) 1693–1711. [38] W.P. Orrick, B.G. Nickel, A.J. Guttmann, J.H.H. Perk, The susceptibility of the square lattice Ising model: new developments, J. Stat. Phys. 102 (2001) 795–841. [39] W.P. Orrick, B.G. Nickel, A.J. Guttmann, J.H.H. Perk, Critical behavior of the two-dimensional Ising susceptibility, Phys. Rev. Letts. 86 (2001) 4120–4123. [40] H. Au-Yang and J.H.H. Perk, Correlation functions and susceptibility in the Zinvariant Ising model, in MathPhys Odyssey 2001: Integrable models and beyond M. Kashiwara and T. Miwa, eds (Birkh¨ auser, Boston, 2002) 23–48

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11 The Pfaffian solution of the Ising model The calculation of the free energy of the two-dimensional Ising model in zero magnetic field is one of the most important computations of theoretical physics of the 20th century. It is one of those few problems that is so important that every physicist should have seen the solution during their graduate education. The present chapter is devoted to the explicit presentation of this calculation. Onsager first computed the free energy of the Ising model at H = 0 in 1944. Since then there have been at least four other methods of solution given. We give the chronology of these methods of computation in Table 11.1. Table 11.1 History of the methods of the computation of the two-dimensional Ising model free energy at H = 0.

1944 1949,1964 1952-1963

1978 1982

Onsager [1] Kaufman, Onsager [2, 3] Schultz, Mattis, Lieb [4] Kac, Ward [5] Potts, Ward [6] Hurst, Green [7] Kasteleyn [8–10] Baxter, Enting [11] Baxter [12]

Onsager’s algebra Fermionization Combinatorial

The 399th solution Functional T Q equation

The 1944 paper of Onsager [1] is one of the most inventive computations in 20th century physics. In it Onsager invents the concept of an infinite dimensional loop algebra and uses it to compute the eigenvalues of the transfer matrix of the Ising model. This algebraic method remained almost totally un-understood until 1985 when it was used to analyze the chiral Potts model. We will touch on this method in chapter 15 but there are still aspects of this profound method which remain to be explored. A partial reason why this 1944 method has not been extensively developed is that in 1949 Kaufman [2] reduced the computation of the partition function to a problem involving free fermions. In 1964 the method was further simplified by Schulz, Mattis and Lieb [4] by the use of the Jordan–Wigner and Bogoliubov transformations. This method is more specialized than Onsager’s original method but in compensation for this restriction in generality the same method computes correlation functions as well as partition functions.

Dimers

¿

The last two methods shown in Table 11.1 rely on the use of what is called the star– triangle equation. The original star–triangle equation was found by Onsager [1], [13] in the Ising model, and in the hands of Baxter has been developed into a method of far-reaching power and generality. These methods will be described in chapters 13 and 14. However, the method lacks the simplicity of the fermionization and combinatorial methods and does not lead by itself to computations of the correlation functions. In this chapter we will present the combinatorial solution of the Ising model. This method is, if you will, a geometrization of the fermionic method of Kaufman. Kaufman’s method relies heavily on group theory and operator methods whereas the combinatorial method uses only a few formulas from linear algebra. Furthermore Kaufman’s method uses the technique of transfer matrices which treat the vertical and horizontal interactions in very different ways whereas the combinatorial method explicitly maintains the symmetry between the vertical and horizontal interactions at all states of the computation. Furthermore the combinatorial method is instantly generalizable to inhomogeneous lattices. We will follow the presentation of the The Two Dimensional Ising model [14, chapters 4 and 5]. In section 11.1 we introduce and solve the dimer problem first solved by Kasteleyn in 1960 [8]. In section 11.2 we reduce the Ising model to a dimer problem and solve for the partition function for several different boundary conditions. In section 11.3 we extend the method to compute correlation functions in terms of determinants. The final results and their analysis have already been given in chapter 10.

11.1

Dimers

A dimer is a figure drawn on a lattice which occupies two sites and the bond connecting them. A dimer configuration is said to be allowed if a site is occupied by no more than one dimer. An allowed close packed dimer configuration is an allowed collection of dimers which occupies all the sites of the underlying lattice We will often be interested in splitting the full set of dimers into several classes. For example of a square lattice we may specialize the bonds into vertical and horizontal. Let the number of dimer configurations of N types of bonds with nj bonds of class j be denoted by g(n1 , · · · , nN ). Then the generating function for the number of configurations is Z(z1 , · · · , zN ) =

n1 ,···,nN

g(n1 , · · · , nN )

N

n

zj j

(11.1)

j=1

where the sum is over all allowed close packed dimer configurations. In this section we will compute an explicit formula for this generating function. The Pfaffian of a set of numbers ajk with 1 ≤ j, k ≤ 2N satisfying the antisymmetry condition ajk = −akj , and akk = 0 (11.2) is defined as

¿¿¼

The Pfaffian solution of the Ising model

PfA =

δp ap1 p2 ap3 p4 · · · ap2N −1 p2N

(11.3)

p

where p1 , · · · , p2N is a permutation of the numbers 1, 2, · · · , 2N , the sum all permutations which satisfy the restrictions p2m−1 < p2m p2m−1 < p2m+1

1≤m≤N 1≤m≤N −1

p

is over (11.4) (11.5)

and δp , the parity of the permutation, is +1 if the permutation p is made up of an even number of transpositions and −1 if the permutation is made up of an odd number of transpositions. We note that, because of (11.2), PfA may be written in the alternative form 1 PfA = δp ap1 p2 ap3 p4 · · · ap2N −1 p2N (11.6) N !2N p where the sum is over all permutations. For 2N = 4 we explicitly have PfA = a12 a34 − a13 a24 + a14 a23 .

(11.7)

For an arbitrary value of N the number of terms in the Pfaffian is (2N − 1)(2N − 3)(2N − 5) · · · 5 · 3 · 1 = (2N − 1)!!.

(11.8)

The usefulness of the Pfaffian stems from the formula [PfA]2 = detA

(11.9)

(where det A is the determinant of the antisymmetric matrix A). Therefore Pfaffians may be studied by the methods of linear algebra. We will demonstrate in 11.1.1 that the generating function for close packed dimers on a planar lattice with free boundary conditions can be written as the Pfaffian of a suitably chosen matrix A and in 11.1.2 the result is extended to cylindrical lattices. In 11.1.3 we extend our considerations to planar lattices of genus g and on these lattices the generating function can be written as the sum of the Pfaffians of 22g different matrices. Proofs for arbitrary planar lattices are given and for concreteness the results are specifically illustrated for the square lattice In 11.1.4 we show that for square lattices with free, cylindrical and periodic boundary conditions the resulting Pfaffians may be explicitly evaluated in terms of finite products using (11.9) and in 11.1.5 the thermodynamic limit is explicitly computed. We conclude in 11.1.6 with a survey of explicit results for other cases such as the triangular lattice and the square lattice with Moebius strip and Klein bottle boundary conditions. 11.1.1

Dimers on lattices with free boundary conditions

We begin the discussion of reducing the generating function of closest packed dimers to the evaluation of a Pfaffian by considering planar lattices with free boundary conditions. It will be most important that this derivation will be valid for any planar

Dimers

¿¿½

lattice and not just for a square or triangular lattice because more general lattices will be needed for the application to the Ising model in section 11.2. Thus, while for concreteness we will use the square lattice to illustrate the method all theorems will be stated in a form which applies to the arbitrary planar lattice. We trust that the reader will see the generalization to arbitrary lattices without the need of the introduction of a general (and cumbersome) notation. We have previously labeled the sites of a square lattice by giving the number of the row j and column k where 1 ≤ j ≤ Lv and 1 ≤ k ≤ Lh . However they may also be labeled by a single index using the map (j, k) ↔ p = k + (j − 1)Lh .

(11.10)

We shall refer to the arrangement of dimers which occupies the pairs of sites p1 and p2 , p3 and p4 , · · · , pLv Lh −1 and pLv Lh as C = |p1 , p2 |p3 , p4 | · · · |pLv Lh −1 , pLv Lh |.

(11.11)

For example the dimer configuration shown in Fig. 11.1 is specified by C0 = |1, 2|3, 4|5, 6| · · · |Lv Lh − 1, Lv Lh |

(11.12)

Fig. 11.1 The configuration C0 .

Suppose we let the nonzero elements of ap1 ,p2 be ap1 ,p2 = zi

(11.13)

where p1 < p2 and p1 and p2 are connected by a bond of class i. As an example for a square lattice, if the classes of bonds are vertical and horizontal zv refers to the vertical and zh refers to the horizontal bonds. It is then obvious that we can write the generating function for closest packed dimers with free boundary conditions ZLFv ,Lh as ZLFv ,Lh =

ap1 p2 ap3 p4 · · · apLv L

p h −1 Lv Lh

(11.14)

p

where the summation is over all permutations satisfying (11.4) and (11.5). This expression is called a Haffnian but, because there is no connection with a determinant, there is no efficient way to study this expression when Lv and Lh become large.

¿¿¾


The expression (11.14) would be a Pfaffian (11.3) if in each term in (11.14) there were an extra factor of δp . Then we could use (11.9) to aid in evaluating the generating function for large Lv , Lh . Therefore we will investigate the possibility that instead of (11.13) we can choose the nonzero elements of ap1 p2 to satisfy

where |s(p1 , p2 )| = 1 and

ap1 p2 = s(p1 , p2 )zi

(11.15)

s(p1 , p2 ) = −s(p2 , p1 )

(11.16)

such that ZLFv ,Lh = PfA =

δP ap1 p2 ap3 p4 · · · apLv Lh −1 pLv Lh .

(11.17)

p

For (11.17) to hold we need to show that if p(1) and p(2) are any two permutations satisfying (11.4) and (11.5) that (1)

(1)

(1)

(1)

(2)

(2)

(2)

(2)

(1)

(1)

(1)

(1)

(2)

(2)

(2)

(2)

δp(1) s(p1 , p2 ) · · · s(pLv Lh −1 , pLv Lh ) = δp(2) s(p1 , p2 ) · · · s(pLv Lh −1 , pLv Lh ). (11.18) However, the restrictions (11.4) and (11.5) are awkward, and therefore it is useful to note that if we let p¯ be any one of the permutations which, by violating (11.4) and (11.5) may be obtained from a given permutation which does satisfy (11.4) and (11.5) then because of (11.16) we see that (11.18) will hold if we can find one such p¯(1) and p¯(2) such that p1 , p¯2 ) · · · s(¯ pLv Lh −1 , pLv Lh ) = δp¯(2) s(¯ p1 , p¯2 ) · · · s(¯ pLv Lh −1 , p¯Lv Lh ). δp¯(1) s(¯ (11.19) With these definitions we proceed to show that for any permutation p there is (at least) one related permutation p¯ for which δp may be computed from geometric considerations. Consider any two arrangements of dimers specified by permutations p(1) and p(2) . Draw the dimers of p(1) on the lattice as dotted lines and the dimers of p(2) as solid lines. The resulting set of figures is referred to as a transition graph. An example is given in Fig. 11.2.

Fig. 11.2 An example of a transition graph.

Dimers

¿¿¿

Every lattice point is the endpoint of one and only one line of each type of dimer and therefore the figures in the transition graph are of two possible types: 1) Two sites connected by a dotted and by a solid line (these figures are called double bonds) and 2) closed polygons with an even number of bonds in which the dotted and solid lines alternate. We call such closed polygons transition cycles because if the bonds of p(1) are permuted clockwise or counterclockwise one step around this cycle they go over to bonds of p(2) . First consider a case in which two permutations differ from each other by only one transition cycle. The preceding discussion showed that (11.18) holds if (11.19) holds where we replace p(1) and p(2) which do obey (11.4) and (11.5) by equivalent permutations p¯(1) and p¯(2) which do not obey (11.4) and (11.5). In particular (11.19) will guarantee (11.18) if we replace p(1) and p(2) by those p¯(1) and p¯(1) that arrange the sites in a clockwise order as we go around the graph. For example consider the transition cycle in Fig. 11.3. 3

4

1

2

Fig. 11.3 A transition cycle of four vertices.

The permutation p(1) obeying (11.4) and (11.5) which specifies the solid dimers is |1, 2|3, 4| and the permutation specifying the dashed dimers is |1, 3|2, 4|. However, a permutation p¯(1) which arranges the sites of the configuration p(1) in clockwise order is |2, 1|3, 4|. Similarly a permutation p¯(2) which arranges the sites of p(2) in clockwise order is |1, 3|4, 2|. Thus the shift from p¯(1) to p¯(2) is a cyclic permutation of one step of four objects. More generally the same argument shows that for any transition cycle there are permutations p¯(1) and p¯(2) such that the shift from one to the other is a cyclic permutation of one step of an even number of objects. Hence this shift involves an odd number of transpositions and accordingly if there is only one transition cycle δp¯(1) = −δp¯(2) .

(11.20)

In general, if there are t transition cycles, we apply this argument one cycle at a time and find that δ (1) = (−1)t δp¯(2) .

(11.21)

Therefore the requirement that the terms associated with the two permutations p(1) and p(2) (or equivalently p¯(1) and p¯(2) ) have the same sign will be satisfied if for each transition cycle (1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

s(¯ p1 , p¯2 )s(¯ p3 , p¯4 ) · · · s(¯ p2N −1 , p¯2N ) = −s(¯ p2 , p¯3 )s(¯ p4 , p¯5 ) · · · s(¯ p2N , p¯1 ). (11.22) Thus for each transition cycle

¿¿

The Pfaffian solution of the Ising model 2N

(1)

(1)

s(¯ pk , p¯k+1 ) = −1

(11.23)

k=1

or, in other words the product of the factors s(p1 , p1 ) in any transition cycle must be −1. It is possible to satisfy (11.23) with complex numbers [15] but in order to obtain the most geometric picture possible we restrict our attention s(p1 , p2 ) = ±1. Then we can represent s(p1 , p2 ) by drawing an arrow on the lattice such that if s(p1 , p2 ) = +1 an arrow points from site p1 to site p2 and s(p1 , p2 ) = −1 an arrow points from site p2 to site p1 . This geometric picture is consistent with the antisymmetry of s(p1 , p2 ). A lattice on which these arrows are drawn will be called an oriented lattice. We define the orientation parity of a transition cycle to be +1 (−1) if, as we traverse this cycle in either direction the number of arrows pointing in the direction of motion is even (odd). The previous discussion therefore proves Theorem A: If the orientation parity of every transition cycle is odd, all terms in the Pfaffian will have the same sign. It is not possible to draw arrows on a general planar lattice so that the orientation parity of every polygon with an even number of sides is odd. However, not all polygons with an even number of sides are transition cycles. To obtain a characterization of transition cycles on a lattice with free boundary conditions it is useful to introduce the concept of inside and outside. For the square lattice drawn in its “natural” configuration Fig 11.4(a) this concept is obvious. However this “natural” definition of inside and outside is not topologically invariant as is demonstrated by the topologically equivalent form of the square lattice shown in Fig. 11.4(b). Our informal definition can be made mathematically unambiguous [16] but for our purposes this degree of rigor is not necessary.

(a)

(b)

Fig. 11.4 (a) The square lattice in its “natural” configuration. The outlined square has no points or bonds of the lattice in its interior. (b) A lattice which is topologically equivalent to the lattice in (a) where the same outlined square now contains many sites and bonds.

Using this concept of inside and outside we may now easily characterize transition cycles. The only figures that occur in transition graphs are double bonds and transition

Dimers

¿¿

cycles. Both of these figures have an even number of sites and they completely cover the lattice as illustrated in Fig. 11.2. Therefore we have: Theorem B: The number of points contained within any transition cycle on a planar lattice is even. To use the property of transition cycles given in theorem B to choose a set of arrows on a planar lattice to satisfy theorem A we make the following definitions: An elementary polygon is a polygon drawn on the lattice (in its natural configuration) which has no points in its interior. An elementary polygon is said to be clockwise odd (even) if the number of arrows pointing in the clockwise direction is odd (even). This even/odd property will be referred to as the orientation parity, and a lattice with arrows drawn on the bonds will be called an oriented lattice. Note that elementary polygons may have either an even or an odd number of sides and that elementary polygons with an odd number of sides which are clockwise odd (even) are counterclockwise even (odd). With these definitions we now state: Theorem C: On any planar lattice (in its “natural” configuration) we always choose a set of arrows such that every elementary polygon is clockwise odd. We will not prove this theorem in general but will rather verify it for the cases of interest. In particular in Fig. 11.5 we exhibit an oriented lattice for the square lattice. It is elementary to verify that the arrows on this lattice satisfy the condition of theorem C. However, we note that this orientation is not unique and that many different orientations exist.

Fig. 11.5 An oriented square lattice.

It is now easy to see that the property of transition cycles of theorem B guarantees that the oriented lattices of Fig. 11.5 satisfy the conditions of theorem A. More generally the proof that theorem B guarantees, for all planar lattices with free boundary conditions, that the orientations specified by theorem C satisfy the conditions of theorem A follows from: Theorem D: Once arrows have been specified such that every elementary polygon

¿¿


is clockwise odd, then for any polygon the number of clockwise bonds is odd if the number of enclosed lattice points is even and is even if the number of enclosed lattice points is odd. We prove theorem D by first remarking that any polygon on the lattice is made up of elementary polygons and that theorem C guarantees that theorem D holds on these elementary polygons. Therefore theorem D will follow by induction if we assume it to be true on all polygons made up of n elementary polygons and prove that if Γn is one of those polygons then the theorem also holds on the polygon Γn+1 obtained by enlarging Γn to include any adjacent elementary polygon Γ1 . Suppose that Γn surrounds p lattice points, contains a clockwise arrows, and has c arrows in common with Γ1 and that the polygon Γ1 contains a clockwise arrows, which from theorem C must be odd. The number of clockwise arrows in Γn+1 is the number of clockwise arrows in Γn plus the number of clockwise arrows in Γ1 minus the number of clockwise arrows lost from Γn , and from Γ1 by omitting common arrows. Now if an arrow on a common bond is clockwise for Γn it will be counter clockwise for Γ1 because Γ1 is outside of Γn . Therefore, the number of clockwise arrows in Γn+1 is a + a − c. Furthermore, the number of enclosed lattice points in Γn+1 is p + c − 1, since when we omit c common bonds we must gain c − 1 new points in the interior. Now by assumption a is even (odd) if p is odd (even) and also a is odd. Therefore a + a − c must be even (odd) if p + c − 1 is odd (even). Hence theorem D follows by induction. The proof of this theorem is illustrated in Fig. 11.6 for the square lattice, but the proof given above is valid for the general case as well. Γ1 Γ7

(a)

Γ1

Γ4

(b)

Γ5

Γ1

(c)

Fig. 11.6 Special cases of the proof of theorem D where on a square lattice an elementary polygon Γ1 is added to a polygon Γn to produce the polygon Γn+1 .

By combining theorems A–D we have now proven that, for any planar lattice with free boundary conditions, the factors s(p1 , p2 ) in (11.15) may be chosen such that all terms in (11.17) have the same sign and thus that ZLFv ,Lh = ±PfAF .

(11.24)

The sign ± must be chosen by the requirement that ZLFv ,Lh be positive when the weights zi are positive. For the square (and triangular) lattice this can be done by noting that the configuration C0 of (11.12) shown in Fig 11.1 has a positive sign. Thus we find for the square lattice with free boundary conditions that ZLFv ,Lh = PfAF

(11.25)

Dimers

¿¿

where the matrix A is determined from (11.15) and the orientation lattices of Fig. 11.5. It remains to explicitly write the matrix AF for the square lattice where we separate the bonds into the classes vertical and horizontal for which we use the weight variables zv and zh respectively. To write the matrix we return to the convention of specifying row and column indices by the pair (j, k) Thus from (11.15) and the orientation lattice of Fig. 11.5 we have a(j, k; j, k + 1) = −a(j, k + 1; j, k) = zh

11.1.2

for 1 ≤ j ≤ Lv , 1 ≤ k ≤ Lh − 1 a(j, k; j + 1, k) = −a(j + 1, k; j, k) = (−1)k+1 zv

(11.26)

for 1 ≤ j ≤ Lv − 1, 1 ≤ k ≤ Lh .

(11.27)

Dimers on a cylinder

The considerations of the previous subsection can readily be extended from free to cylindrical boundary conditions. For concreteness we consider the case where locally the lattice is square or triangular and impose cylindrical boundary conditions in the horizontal direction. There are then two cases to consider depending on whether Lh is even or odd. Lh odd As in the previous section we may consider reducing the generating function of close packed dimers to a Pfaffian by considering transition cycles between two dimer configurations specified by any two permutations p(1) and p(2) . However when Lh is odd there can be no transition cycle which completely loops the lattice and therefore the considerations of the previous section guarantee that for the square lattice the generating function for closest packed dimer configurations on the cylindrical lattice is ZLc v ,Lh = PfAc

(11.28)

where the elements of the matrices Ac are given by (11.26) and (11.27) and by the additional nonzero elements a(j, Lh ; j, 1) = −a(j, 1; j, Lh ) = zh

for 1 ≤ j ≤ Lv

(11.29)

Lh even Now there will be two distinct classes of transition cycles: 1) those which do not loop the cylinder and 2) those which loop the cylinder precisely once. One particularly simple class two transition cycle has no vertical bonds and is shown in Fig. 11.7. There are, in addition, Lv − 1 transition cycles which differ from this one only by a vertical translation. These Lv transition cycles will be called elementary transition cycles of class two. The class one transition cycles will be properly included in the Pfaffian if the matrix A is given as in the previous case by (11.26), (11.27) and (11.29). However, this specification of arrows is not unique and it is easily verified that because all class

¿¿


Fig. 11.7 An elementary class two transition cycle on a cylinder.

one transition cycles must have an even number of bonds which connect column Lh with column 1 that the matrix given by (11.26), (11.27) and a(j, Lh ; j, 1) = −a(j, 1; j, Lh ) = −zh

for 1 ≤ j ≤ Lv

(11.30)

works just as well. However, class two transition cycles have only one bond between column Lh and column 1 and thus the orientation parity of an elementary class 2 transition cycle is negative only with the choice (11.30). Furthermore it is straightforward to demonstrate that if class one transition cycles and elementary class two transition cycles are counted properly that all class two transition cycles are counted properly. Therefore we have shown that (11.28) holds with Ac given by (11.26), (11.27) and (11.30). 11.1.3

Dimers on lattices of genus g ≥ 1

The final type of boundary conditions we need to investigate are toroidal boundary conditions where the lattice is periodic in both the vertical and horizontal directions. This case proves to be at the same time both harder and easier than free or cyclic boundary conditions. It is easier because the periodic boundary conditions allow a simple evaluation of Pfaffians and determinants by means of Fourier transforms. It is harder because for toroidal boundary conditions the generating function is no longer one Pfaffian but is instead the sum of four Pfaffians. We will present the four Pfaffian formula in this subsection and explicitly evaluate the related determinants in 11.1.4. However, our methods are more general than merely dealing with a torus which has genus 1 but will also be applicable to boundary conditions which can be embedded on a surface of genus g in which case the resulting sum of Pfaffians will have 22g terms. This generalization was noted in the original articles [9, 10] which derived the results for the toroidal case but until rather recently [16, 17] they have not been greatly studied because 1) no one has discovered how to explicitly evaluate the Pfaffians and 2) they are of marginal interest in statistical mechanics. However, the identical problem has more recently arisen in contexts other than statistical mechanics where surfaces of higher genus are of physical interest and thus we will state the general results for future reference. For lattices with toroidal boundary conditions we now may have, in addition to transition cycles which do not loop the lattice, transition cycles which loop the torus v

Dimers

¿¿

times in the vertical direction and h times in the horizontal direction where is it easily seen that h and v must be relatively prime (if they are not both zero). It is a simple matter to extend the considerations of 11.1.1 and 11.1.2 to ensure that all dimer configurations connected by transitions cycles with v = h = 0 are counted with the same sign. Indeed, there are many ways to do this and in particular we will need the following four different matrices A±± where the elements a(j, k; j , k )±,± are given by (11.26) and (11.27) for all bonds which do not connect row Lv with row 1 or column Lh with column 1. For the rest of the elements we have the following a±± (j, Lh ; j, 1) = −a±± (j, 1; j, Lh ) = ±zh 1 ≤ j ≤ Lv a±± (Lv , k; 1, k) = −a±± (1, k; Lv , k) = ± (−1)k+1 zv 1 ≤ k ≤ Lh .

(11.31)

Unlike the case of cylindrical boundary conditions the Pfaffians of none of the matrices A±± will count all configurations with the same sign. These signs may be efficiently determined by comparing any particular dimer configuration with the configuration C0 (11.12) of Fig. 11.1. Transition cycles where one of the dimer configurations is C0 are called C0 transition cycles and to correctly compute the generating function it is sufficient to include all C0 transition cycles with the same sign. It is shown in [9], [10], and [14] that for v and h relatively prime that the Pfaffians of the four matrices A±± include C0 transition cycles with the signs shown in Table 11.2. Table 11.2 Signs with which C0 transition cycles are included in the Pfaffian of A±± .

(h, v) (0, 0) (odd,even) (even,odd) (odd,odd)

A++ + − − −

A+− + − + +

A−+ + + − +

A−− + + + −

Thus although none of the four matrices includes all C0 transition cycles with the same sign there is a linear combination of the four Pfaffians which will correctly give the generating function ZLPv ,Lh =

1 {−PfA++ + PfA+− + PfA−+ + PfA−− }. 2

(11.32)

The result (11.32) can be extended to genus g > 1 surfaces there are 22g matrices analogous to (11.31) with one matrix for each of the 22g ways that ± signs can be assigned to the 2g loops of the surface and there is a linear combination of Pfaffians which correctly gives the generating function [9, 10]. Explicit examples for g = 2 given in [17]. 11.1.4

Explicit evaluation of the Pfaffians

Strictly speaking, if all we are interested in is the Ising model we do not need to pursue the dimer problem further, However, because dimers are interesting as a statistical problem in their own right and because the techniques used will be needed for the Ising model we will complete the study of the dimer problem by explicitly evaluating the

¿


Pfaffians by means of the determinental formula (11.9). We proceed in the opposite order from that used in the derivation of the generating functions in terms of the Pfaffians and consider toroidal (periodic) boundary conditions first, and cyclic and free boundary conditions later. Toroidal (periodic) boundary conditions The four matrices A±,± (with Lh even) are written in a compact form by defining ± four N × N matrices, IN the identity matrix, FN (with N even) and JN where 

FN

  1 0 0 0 ··· 0 0 1  0 −1 0 0 · · · 0   −1 0    0 0 1 0 ··· 0    ± =  0 0 0 −1 · · · 0  , JN =  0 −1  .. ..    .. .. .. .. .. ..   . .  . . . . . . ±1 0 0 0 0 0 · · · −1

 0 ∓1 0 0  0 0  .. ..  . . 0 · · · 0 −1 0 0 ··· 0 1 ··· 0 0 ··· 0 .. .. .. . . .

(11.33)

Then the matrices A±± are written in a direct product notation as

A±± = zh ILv ⊗ JL±h + zv JL±v ⊗ FLh

(11.34)

where the labeling of the basis is such that

a±± (j, k; j , k ) = zh [ILv ]j,j [JL±h ]k,k + zv [JL±v ]j,j [FLh ]k,k .

(11.35)

To proceed further it is convenient to define for even N one more N × N matrix:   i 0 0 0 ··· 0 0 1 0 0 ··· 0   0 0 i 0 ··· 0   TN =  0 0 0 1 · · · 0  (11.36)    .. .. .. .. .. ..  . . . . . .  0 0 0 0 ··· 1 Then, multiplying A±± on the right by ILv ⊗ TLh and on the left by ILv ⊗ (−iTLh ), and calling the resulting matrix A¯±± we use det[ILv ⊗ TLh ] = (i)Lv Lh /2 ,

det[ILv ⊗ (−iTLh )] = (−i)Lv Lh /2

(11.37)

to obtain where

detA±± = detA¯±±

(11.38)

A¯±± = zh ILv ⊗ JL±h + izv JL±v ⊗ ILh .

(11.39)

± Let λk with k = 1, · · · , N be the N eigenvalues of JN . Then the Lv Lh eigen(L ;±) (L ;±) v h values of A¯1 are zh λk + izv λj where 1 ≤ j ≤ Lv and 1 ≤ k ≤ Lh . Then (N ;±)

Dimers

detA

±±

Lv Lh

=

(Lh ;±)

[zh λk

(Lv ;± )

+ izv λj

].

¿

(11.40)

j=1 k=1 ± ± It remains to calculate the eigenvalues of JN . We note that JN has two important properties. First, the matrix elements are of the form

aj,k = aj−k .

(11.41)

+ that if the first row Matrices of this form are called Toeplitz. Second, we note for JN is transposed to the bottom of the matrix and the first column is transposed to the extreme right of the matrix the matrix is transformed into itself. Toeplitz matrices of − this form are called cyclic. Correspondingly JN is called near cyclic. The eigenvalues of (near) cyclic matrices are easily evaluated. For the specific case ± at hand eigenvalue equation JN v = λv is explicitly written in component form as

v2 ∓ vN = λv1 −vk−1 + vk+1 = λvk for 2 ≤ k ≤ N − 1 ±v1 − vN −1 = λvN .

(11.42) (11.43) (11.44)

We seek a solution of the form vk = αk

(11.45)

and then find that the N equations in (11.42)–(11.44) reduce to the three equations α ∓ αN −1 = λ

(11.46)

−1

−α + α = λ ±α−N +1 − α−1 = λ

(11.47) (11.48)

and these three equations are identical if αN = ±1.

(11.49)

Therefore the N eigenvectors with 1 ≤ n ≤ N are (n;+)

vk

(n;−) vk

= e2πikn/N =e

(11.50)

πik(2n−1)/N

(11.51)

and the corresponding eigenvalues as λn(N ;+) = e2πin/N − e−2πin/N = 2i sin(2πn/N ) λn(N ;−)

=e

πi(2n−1)/N

−e

−πi(2n−1)/N

= 2i sin(π(2n − 1)/N ).

(11.52) (11.53)

Using (11.53) in (11.40) we obtain detA++ =

Lh Lv j=1 k=1

2πk 2πj 2izh sin − 2zv sin Lh Lv

(11.54)

¿


detA+− =

Lv Lh j=1 k=1

detA−+ =

Lv Lh

j=1 k=1

detA−− =

Lv Lh

j=1 k=1

2πk π(2j − 1) 2izh sin − 2zv sin Lh Lv π(2k − 1) 2πj 2izh sin − 2zv sin Lh Lv

(11.55)

(11.56)

π(2k − 1) π(2j − 1) 2izh sin . − 2zv sin Lh Lv

(11.57)

In A++ the term in the product with j = Lv and k = Lh vanishes for all zv and zh and hence (11.58) PfA++ = (detA++ )1/2 = 0. Thus using PfA = ±(detA)1/2

(11.59)

and (11.55)–(11.57) in (11.32), choosing the signs in (11.59) to make the three nonzero Pfaffians positive for zv , zh > 0 and using the fact that Lh is even by definition we obtain the final result ZLPv ,Lh

1/2 Lv L h /2 1 2 2πk 2 π(2j − 1) 2 2 4zh sin = { + 4zv sin 2 j=1 Lh Lv k=1

+

Lv L h /2

π(2k − 1) 2πj + 4zv2 sin2 Lh Lv

4zh2 sin2

π(2k − 1) π(2j − 1) + 4zv2 sin2 Lh Lv

j=1 k=1

+

Lv L h /2

1/2

4zh2 sin2

j=1 k=1

1/2 }. (11.60)

The square roots in this expression are only apparent because, in each of the three terms, the factors in the bracket in the double products are either perfect squares or they occur in pairs. Free and cylindrical boundary conditions To explicitly evaluate the Pfaffians of AF and Ac for the generating functions of free (11.25) and cylindrical (11.28) boundary conditions we define one further N × N matrix   0 1 0 ··· 0 0  −1 0 1 · · · 0 0      (11.61) JN =  0 −1 0 · · · 0 0   .. .. .. .. .. ..   . . . . . . 0

0 0 · · · −1 0

and write Ac = zh ILv ⊗ JL−h + zv JL v ⊗ FLh ,

AF = zh ILv ⊗ JL h + zv JL v ⊗ FLh . (11.62)

Dimers

¿

As for the case of the previous subsection we compute the determinants of these matrices by first multiplying on the right by ILv ⊗TLh and on the left by ILv ⊗(−iTLh ). (N ; ) Thus denoting the eigenvalues of JN by λk we have detAc =

Lh Lv

(Lh ;−)

(zh λk

(Lv ; )

+ izv λj

), detAF =

j=1 k=1

Lh Lv

(Lh ; )

(zh λk

(Lv ; )

+ izv λj

).

j=1 k=1

(11.63) are obtained by writing the eigenvalue equation in compoThe eigenvalues of JN nent form v2 = λv1 −vk−1 + vk+1 = λvk for 2 ≤ k ≤ N − 1 −vN −1 = λvN

(11.64)

which is equivalent to −vk−1 + vk+1 = λvk for 1 ≤ k ≤ N

(11.65)

with the boundary conditions v0 = vN +1 = 0.

(11.66)

The most general solution to the second-order difference equation (11.65) is vk = Aαk+ + Bαk−

(11.67)

α − α−1 = λ.

(11.68)

where α+ and α− satisfy Therefore α± =

1 [λ ± (λ2 + 4)1/2 ] 2

(11.69)

α+ α− = −1.

(11.70)

and in particular To satisfy the boundary conditions (11.66) we need v0 = A + B = 0 +1 +1 + BαN = 0. vN +1 = AαN + −

(11.71) (11.72)

From (11.70), (11.71) and (11.72) we have 2(N +1)

α+

= (−1)N +1

and thus we find for 1 ≤ n ≤ 2(N + 1) ∓exp ∓πi(2n−1) 2(N +1) N even α± = ∓exp ∓πin N odd. N +1

(11.73)

(11.74)

¿


; so the eigenvalues λN are contained in the set for 1 ≤ n ≤ 2(N + 1) n

2i sin

πn π(2n − 1) , for N even, 2i sin , for N odd. 2(N + 1) N +1

(11.75)

From (11.69) we see that each λ = 2i has two distinct eigenvectors. Each eigenvalue is counted twice in the set (11.75) and therefore we extract a nondegenerate set by setting for 1 ≤ n ≤ N + 1 N +2 n + 2 , N even n= (11.76) n + N2+1 , N odd. However, if n = N +1 then λ = −2i and this value of λ has only the trivial eigenvector vk = 0. Therefore we find the desired result

; λN n = 2i cos

πn , N +1

1 ≤ n ≤ N.

(11.77)

We now use (11.53) and (11.77) in (11.63) and choose the sign of the square root to make the Pfaffians in (11.25) and (11.28) positive to obtain the desired results for Lv and Lh even 4zh2 sin2

π(2k − 1) πj (11.78) + zv2 cos2 Lh Lv + 1

4zh2 cos2

πk πj + zv2 cos2 Lh + 1 Lv + 1

Lh /2 Lv /2

ZLc v ,Lh = [detAc ]1/2 =

k=1 j=1 Lh /2 Lv /2

ZLFv ,Lh

c 1/2

= [detA ]

=

k=1 j=1

(11.79)

and for Lv odd and Lh even

Lh /2

ZLc v ,Lh

=

k=1

π(2k − 1) 2zh sin Lh

(Lv −1)/2

4zh2 sin2

j=1

π(2k − 1) πj + zv2 cos2 Lh Lv + 1

(11.80)

Lh /2

ZLFv ,Lh =

k=1

11.1.5

2zh cos

πk Lh + 1

(Lv −1)/2

j=1

4zh2 cos2

πk πj + zv2 cos2 . (11.81) Lh + 1 Lv + 1

Thermodynamic limit

It remains to compute the thermodynamic limit Lv , Lh → ∞ of the dimer generating functions computed above. The products may all be written as the exponential of the sum of the logarithms and to leading order the sums may be replaced by integrals in the thermodynamic limit. The result is the same for all boundary conditions and thus we find that

Dimers

¿

fdSQ = lim (Lv Lh )−1 ln ZLFv ,Lh Lv →∞ Lh →∞

= lim (Lv Lh )−1 ln ZLc v ,Lh = lim (Lv Lh )−1 ln ZLPv ,Lh Lv →∞ Lh →∞

= (2π)−2

π

π

dθ2 ln[4(zv2 sin2 θ1 + zh2 sin2 θ2 ].

dθ1 0

Lv →∞ Lh →∞

(11.82)

0

However, if we expand these generating functions beyond leading order they are no longer equal. In particular, the generating function for toroidal boundary conditions, which on the lattice consists of three separate terms, has been more accurately expanded [18] as 2 2 2 θ2 (0|τ ) 1 θ3 (0|τ ) θ4 (0|τ ) P ZLv ,Lh = exp(fdSQ Lv Lh ) + + (11.83) 2 η(τ ) η(τ ) η(τ ) where η(τ ) and the theta functions of zero argument θi (0|τ ) are defined by (10.37)– (10.40), the modulus is τ = iLv zh /Lh zv (11.84) and multiplicative corrections are of order (ln Lv Lh )3 /Lv Lh . 11.1.6

Other lattices and boundary conditions

We have illustrated the general theory of close packed dimer statistics with the explicit computation for the square lattice. However, many other problems have been solved, and we conclude this section by surveying the results. Triangular lattice The Pfaffian for the triangular lattice with free boundary conditions is obtained from the orientation lattice of Fig. 11.8.

Fig. 11.8 The orientation lattice for the triangular lattice with free boundary conditions.

Cylindrical and toroidal boundary conditions are treated exactly as for the square lattice. The generating function for the toroidal lattice is computed to be [19]

¿


ZLPvT,Lh =

1/2 Lv L h /2 2πk 2πj 2πk 2πj 1 {− 4zh2 sin2 + 4zv2 sin2 + 4zd2 cos2 + 2 Lh Lv Lh Lv j=1 k=1

Lh /2

+

Lv

4zh2 sin2

2πk π(2j − 1) + 4zv2 sin2 + 4zd2 cos2 Lh Lv

4zh2 sin2

π(2k − 1) 2πj + 4zv2 sin2 + 4zd2 cos2 Lh Lv

j=1 k=1

+

Lv L h /2 j=1 k=1

+

Lv L h /2 j=1

2πk π(2j − 1) + Lh Lv π(2k − 1) 2πj + Lh Lv

π(2k − 1) π(2j − 1) + 4zv2 sin2 Lh Lv k=1 1/2 π(2k − 1) π(2j − 1) +4zd2 cos2 + }. Lh Lv

1/2 1/2

4zh2 sin2

(11.85)

This differs from the corresponding result for the square lattice (11.60) in that now all four Pfaffians are nonzero. In the thermodynamic limit the leading term in the generating function is the same for all boundary conditions and thus for the triangular lattice fdT = lim (Lv Lh )−1 ln ZLFv ,Lh Lv →∞ Lh →∞

= lim (Lv Lh )−1 ln ZLc v ,Lh = lim (Lv Lh )−1 ln ZLPv ,Lh Lv →∞ Lh →∞

= (2π)−2

π

π

dθ2 ln[4(zv2 sin2 θ1 + zh2 sin2 θ2 + zd2 cos(θ1 + θ2 )]. (11.86)

dθ1 0

Lv →∞ Lh →∞

0

A more accurate expansion for Lv , Lh → ∞ for toroidal boundary conditions with zv , zh , zd > 0 is T ZLTv ,Lh = efb Lv Lh (1 + O(econstLv Lh )) (11.87) which does not have the theta function terms of the result (11.83) for the square lattice with zd = 0. Moebius strips and Klein bottles The discussion of boundary conditions of the previous sections can be extended to lattices on nonorientable surfaces [16]. The Moebius strip has boundary conditions twisted as shown on the left in Fig. 11.9 where the horizontal bonds at the end of the lattice are identified as shown. A Klein bottle is obtained by imposing periodic boundary conditions on the free boundary of the Moebius strip as shown on the right in Fig. 11.9. The generating function for dimer configurations on both these lattices is given as a single Pfaffian [16, 20]. The Pfaffians may be explicitly evaluated [20] as

1

The Ising partition function 3 4 5 6

2

1

6

1

6

2

5

2

5

3

4

3

4

4

3

4

3

5

2

5

2

6

1

6

1 1

2

3

(a)

4

5

¿

6

(b)

Fig. 11.9 Moebius strip (a) and Klein bottle (b) boundary conditions on a square lattice.

 ZLv ,Lh =

L L /2 zh v h Re (1

− i)

[Lv /2] Lh j=1 k=1

 π(4k − 1) 2i(−1)Lv /2+j+1 sin + Xj  2Lh (11.88)

where Xj =

11.2

2(zv /zh ) cos Lπj for the Moebius strip v +1 π(2j−1) 2(zv /zh ) sin Lv for the Klein bottle.

(11.89)

The Ising partition function

We now turn our attention to the topic of principal interest, the computation of the partition function for the Ising model at H = 0 defined by the interaction energy E = −E h

Lh Lv

σj,k σj,k+1 − E v

j=1 k=1

Lh Lv

σj,k σj+1,k

(11.90)

j=1 k=1

where σj,k = ±1. We will solve this problem by reducing it to the computation of the generating function of dimers on a related “counting” lattice. We treat the case of toroidal and cylindrical boundary conditions separately and for cylindrical boundary conditions we will be able to let an external magnetic field interact with the boundary row of spins. 11.2.1

Toroidal (periodic) boundary conditions

We impose periodic boundary conditions by setting σj,Lh +1 ≡ σj,k and σLv +1,k ≡ σ1,k . and write the partition function as   Lv Lv Lh Lh ZLIPv ,Lh = e−βE = exp βE h σj,k σj,k+1 + βE v σj,k σj+1,k  σ=±1

=

σ=±1

 

σ=±1 Lh Lv j=1 k=1

eβE

h

j=1 k=1

 σj,k σj,k+1  

Lh Lv

j=1 k=1

 eβE

v

σj,k σj+1,k 

j=1 k=1

.

(11.91)

¿


Now, since σ may only take on the values ±1, we have βE if σσ = 1 e βEσσ e = = cosh βE + σσ sinh βE. −βE e if σσ = −1

(11.92)

Therefore we may write the partition function as ZLIPv ,Lh = (cosh βE h cosh βE v )Lv Lh

Lh Lv

(1 + zh σj,k σj,k+1 )(1 + zv σj,k σj+1,k )

σ=±1 j=1 k=1

(11.93) where zv = tanh βE v and zh = tanh βE h .

(11.94)

3 We now expand the products over j and k. Any term with a factor σj,k or σj,k 2 vanishes when the sum over σj.k = ±1 is taken. For all other terms we use σj,k = 4 σj,k = 1 and 1 = 2Lv Lh (11.95) all σ=±1

and thus we find ZLIPv ,Lh = (2 cosh βE h cosh βE v )Lv Lh

Np,q zhp zvq

(11.96)

p,q

where Np,q is the number of configurations with the following properties: i) each bond between nearest neighbors can be used at most once; ii) an even number of bonds terminate at each vertex; iii) each figure has p horizontal and q vertical bonds. An example of such a figure on a portion of a square lattice is given in Fig. 11.10.

Fig. 11.10 A polygon figure on a portion of a square lattice which contributes to the Ising partition function of (11.96).


¿

We evaluate this generating function by mapping it to a problem of close packed dimers. This is done by replacing each site of the square lattice where four bonds intersect by the cluster of six sites shown in Fig. 11.11. We call this dimer lattice the “counting lattice”.

Fig. 11.11 The six-site cluster used to convert the Ising problem into a dimer problem.

There is a one to one correspondence between closed polygon configurations on the Ising lattice and close packed dimers on the counting lattice as is verified by the map of Fig. 11.12. ISING

DIMER

ISING

(1)

(5)

(2)

(6)

(3)

(7)

(4)

(8)

DIMER

Fig. 11.12 The one-to-one equivalence of the Ising model vertex configurations to dimer configurations on the six-site cluster.

¿


Therefore if the weights on the six internal sites of the dimer counting lattice are set equal to one and the weights of the original horizontal and vertical bonds are called zh and zv then the generating function for close packed dimers with these weights on the counting lattice is equal to the generation function for polygons on the Ising lattice. Hence we have reduced the computation of the partition function of the Ising model to a problem in dimer statistics. This dimer problem is solved by the methods introduced in the previous section. Because we are studying periodic boundary conditions the generating function is given by the linear combination of four Pfaffians ZLIPv ,Lh =

1 ¯IP ¯IP ¯IP {−Pf A¯IP ++ + Pf A+− + Pf A−+ + Pf A−− } 2

(11.97)

where the matrices A¯IP ±± are 6Lv Lh × 6Lv Lh matrices specified in the interior by the oriented lattice of Fig. 11.13. U

U L

2

L 1 D

2

R

2

L 1

L

L 1

U L

U

1

D

D

U

U

2

R

2

R

L 1

2

R

D

D

1 D

U

U

U

2

L 1 D

R

2

L 1 D

R

R

R

2 1

R

D

Fig. 11.13 A portion of the dimer counting lattice with six-site clusters. The bonds internal to the six-site clusters have weight 1, the vertical bonds between U and D sites are zv and the horizontal bonds between R and L sites are zh .

We label the elements of the matrix by the site (j, k) of the original Ising lattice and the six indices R, L, U, D, 1, 2 for the six sites of the cluster as indicated in Fig. 11.13. Thus we have


A¯IP (j, k; j, k)±±

R L U = D 1 2

 0 0 −1 0 0 1  0 0 0 −1 1 0     1 0 0 0 0 −1     0 1 0 0 −1 0     0 −1 0 1 0 1  −1 0 1 0 −1 0

¿



for

1 ≤ j ≤ Lv 1 ≤ k ≤ Lh

A¯IP (j, k; j, k + 1)±± = −(A¯IP )T (j, k + 1; j, k)±±   R 0 zh 0 0 0 0  L 0 0 0 0 0 0  U 0 0 0 0 0 0  for 1 ≤ j ≤ Lv =  0 0 0 0 0 0 D 1 ≤ k ≤ Lh − 1   1 0 0 0 0 0 0 2 0 0 0 0 0 0 A¯IP (j, k; j + 1, k)±± = −(A¯IP )T (j + 1, k; j, k)±±   R 0 0 0 0 0 0  L 0 0 0 0 0 0  0 0 0 z U  0 0 v   for 1 ≤ j ≤ Lv − 1 =  D 0 0 0 0 0 0 1 ≤ k ≤ Lh  1 0 0 0 0 0 0 2 0 0 0 0 0 0

(11.98)

(11.99)

(11.100)

A¯IP (j, Lh ; j, 1)±± = −(A¯IP )T (j, 1; j, Lh )±± = ±A¯IP (j, 1; j, 2)±± , 1 ≤ j ≤ Lv A¯IP (Lv , k; 1, k)±± = −(A¯IP )T (1, k; Lv , k)±± = ± A¯IP (1, k; 2, k)±± , 1 ≤ k ≤ Lh . (11.101) We will compute the Pfaffians of A¯IP ±± by using ¯IP 1/2 . Pf A¯IP ±± = ±(detA±± )

(11.102)

The dimensions of the matrices A¯IP ±± are 6Lv Lh × 6Lv Lh . However, it is trivially possible to find 4Lv Lh × 4Lv Lh matrices AI±± such that IP 1/2 . Pf A¯IP ±± = ±(detA±± )

(11.103)

This is done by noticing that if in every 6 × 6 submatrix A¯IP (j, k; j k )±± we subtract column 1 from column R, add column 1 to column U , add column 2 to column L and subtract column 2 from column D we have IP detA¯IP ±± = detA±±

where

(11.104)

¿


  0 1 −1 −1 R  L  −1 0 1 −1  (11.105) AIP (j, k; j, k)±± = U  1 −1 0 1  1 1 −1 0 D IP and all other elements A (j, k; j , k )±± are identical to A¯IP (j, k; j , k )±± with the rows and columns labeled 1 and 2 removed. The matrix AIP (j, k; j, k)±± can be given an interpretation in terms of a nonplanar graph as indicated in Fig 11.14. U L

R

D Fig. 11.14 A representation of the matrix AIP (j, k; j, k)±± as an oriented nonplanar graph.

We now turn to the evaluation of detAIP ±± . We do this by first defining   0 1 0 ··· 0  0 0 1 ··· 0     ± HN =  ... ... ... ... ...     0 0 0 ··· 1 ±1 0 0 · · · 0 and then writing AIP ±± in a direct product notation as   0 1 −1 −1  −1 0 1 −1   ⊗ ILv ⊗ IL  AIP ±± =  1 −1 h 0 1 1 1 −1 0    0 zh 0 0 0 0 0 0 0 0  −zh 0 ±   +  0 0 0 0  ⊗ ILv ⊗ HLh +  0 0 0 0 0 0 0 0    0 0 0 0 0 0 0 0 0 0 0  0 0 0 ±   +  0 0 0 zv  ⊗ HLv ⊗ ILh +  0 0 0 0 0 −zv 0 0 0 0

0 0 0 0

(11.106)

 0 0  ⊗ ILv ⊗ H ±T Lh 0 0 

0 0  ⊗ H ± T ⊗ ILh . Lv 0 0

(11.107) +

−

The matrices H (H ) are (near) cyclic matrices and therefore their eigenvalues are computed by the methods of section 11.1 to be e2πin/N and eπi(2n+1)/N respectively ± for n = 1, · · · , N. Furthermore the matrices HN are unitary


H ± H ±T = 1

(11.108)

and therefore the diagonalizing matrix U has the property that if  2πi/N  e 0 ··· 0 0 e4πi/N · · · 0    + −1 U HN U = . .. .. ..   .. . . . 0

+T −1 U U HN

e−2πi/N 0  = .  .. 0

and similarly if



− −1 U HN U

eπi/N 0  = .  .. 0 

then −T −1 U HN U

e−πi/N 0  = .  .. 0

(11.109)

··· 1

0



then

¿

0 e−4πi/N .. .

 ··· 0 ··· 0  .. ..  . .

0

··· 1  0 0  ..  .

0 e3πi/N .. .

··· ··· .. .

0

· · · −1  0 0  ..  . .

0 e−3πi/N .. .

··· ··· .. .

0

· · · −1

(11.110)

(11.111)

(11.112)

Therefore we evaluate by first transforming HL±v and HL±h into diagonal form by a similarity transformation to obtain detAIP detAIP (θh , θv ) (11.113) ±± = detAIP ±±

± θh θv±

where the products over θk+ = 2πn/Lk and θk− = π(2n − 1)/Lk for k = v, h and   0 1 + zh eiθh −1 −1  −1 − zh e−iθh  0 1 −1  AIP (θh , θv ) =  (11.114) iθv  .  1 −1 0 1 + zv e 1 1 −1 − zh e−iθv 0 It is now straightforward to compute the determinant of the 4 × 4 matrix AIP (θh , θv ) and thus we find detAI±± = [(1 + zh2 )(1 + zv2 ) − 2zh (1 − zv2 ) cos θh − 2zv (1 − zh2 ) cos θv ]. (11.115) ± θh θv±

Using this in (11.97) thus completes the exact evaluation of the partition function of the two-dimensional Ising model at H = 0 first obtained by Kaufman [2] which was

¿


presented in chapter 10. For real zv and zh the factors in the product (11.115) are nonnegative and bounded below by (|zv zh | + |zv | + |zh | − 1)2 .

(11.116)

When the temperature is chosen such that zv zh + zv + zh − 1 = 0

(11.117)

there is a vanishing factor in the product over θh+ , θv+ . For the value of the temperature T determined by (11.117) we have detAIP ++ = 0

(11.118)

and the expression for the partition function reduced to the sum of three terms just as it did for dimers on the square lattice. The thermodynamic limit can now be explicitly taken and thus we find the famous result of Onsager [1] presented in chapter 10 that the free energy is −FI /kB T = lim (Lv Lh )−1 ln ZLIPv ,Lh Lv →∞ Lh →∞

2π 2π 1 = ln(2 cosh βE h cosh βE v ) + (2π)−2 dθh dθv ln[(1 + zh2 )(1 + zv2 ) 2 0 0 − 2zh (1 − zv2 ) cos θh − 2zv (1 − zh2 ) cos θv ] 2π 2π 1 = ln 2 + (2π)−2 dθh dθv ln[cosh 2βE h cosh 2βE v 2 0 0 − sinh 2βE h cos θh − sinh 2βE v cos θv ].

(11.119)

This integral fails to be analytic as a function of T at the point determined by (11.117) which is the condition for the critical temperature Tc discussed in chapter 10. 11.2.2

Cylindrical boundary conditions

When, instead of using toroidal boundary conditions, we impose cyclic boundary conditions only in the horizontal direction we can let the boundary row interact with an external magnetic field and the problem is still solvable by dimer methods. Thus we consider E = −E h

Lh Lv j=1 k=1

σj,k σj,k+1 − E v

Lh L v −1

σj,k σj+1,k − Hb

j=1 k=1

Lh

σ1,k

(11.120)

k=1

The methods used above for the toroidal lattice reduce this problem to a dimer problem on the counting lattice of Fig. 11.15 where the magnetic field is included by introducing an extra row of spins at j = 0 with horizontal weights of unity and vertical weights of z = tanh βHb . Thus we find that the partition function is given as a single Pfaffian

(11.121)

Correlation functions

ZlIc = (2 cosh βE h )Lv Lh (cosh βE v )Lh (Lv −1) PfAIc v ,Lh

¿

(11.122)

where the matrix is determined from Fig. 11.15 with the horizontal arrows connecting columns Lh and 1 in the direction opposite to all other horizontal arrows. The evaluation of this Pfaffian was carried out in [21]. The results were presented in chapter 10.

z

1

z

z

1 Ising

1

z

1

1

1 Dimer

Fig. 11.15 The counting lattice for the Ising model on a cylinder with a magnetic field on the boundary row j = 1 and the corresponding oriented dimer lattice with four-site clusters. The magnetic field is included by introducing an extra row of spins at j = 0 with horizontal weights of unity and vertical weights of z = tanh βHb .

11.3


The great virtue which the combinatorial method as presented above has over the original method of Onsager [1] and over the methods based on the star–triangle equation [11, 12] is that, in addition to computing the partition function, we may also obtain expressions for correlation functions of any number of spins in terms of determinants [22]. We illustrate the method in detail for the correlation of two spins in the same row. We also consider the correlation on the diagonal σN ,N σN,N and near the boundary of a cylinder. 11.3.1

The correlation σM,N σM,N

The two-point function σM ,N σM,N is defined by σM ,N σM,N =

1

ZLIcv ,Lh σ=±1

σM ,N σM,N e−βE

(11.123)

where for purposes of illustration we have (temporarily) chosen cylindrical boundary conditions. The exponentials in the numerator of (11.123) may be expanded using (11.92) and thus we find σM ,N σM,N = (cosh βE h )Lh Lv (cosh βE v )Lh (Lv −1) ZLIc−1 v ,Lh σ=±1

σM ,N σM,N

Lh Lv j=1 k=1

(1 + zh σj,k σj,k+1 )

Lh L v −1 j=1 k=1

(1 + zv σj,k σj+1,k ).(11.124)

¿


The key step is now to show that the numerator in (11.124) can be rewritten as a partition function on a new modified lattice which can also be expressed as a Pfaffian. Connect the spins at M, N and M, N by any path on the lattice we wish. (For example we may consider the straight line path which uses the sites M, k with k = N + 1, · · · , N − 1). On each of the sites j, k on the path between σM,N and σM,N 2 insert the factor of 1 = σj,k and associate the factors into nearest neighbor pairs as (for the example of the straight line path) σM,N σM,N = (σM,N σM,N +1 )(σM,N +1 σM,N +2 ) · · · (σM,N −1 σM,N ).

(11.125)

Then we may multiply (11.125) by the factors 1 + zv σj,k σj+1,k and 1 + zh σj,k σj,k+1 in (11.123) which involving the same nearest neighbor pairs using the identity σσ (1 + zσσ ) = z(1 + z −1 σσ ).

(11.126)

Thus for the straight line path we may write the correlation function σM,N σM,N as

σM,N σM,N = (cosh βE h )Lh Lv (cosh βE v )Lh (Lv −1) zhN −N   N −N −1 1 −1  × (1 + zh σM,k σm,k+1 ) ZLIcv ,Lh σ=±1 k=0    L Lh L L v v −1 h (1 + zh σj,k σj,k+1 )  (1 + zv σj,k σj+1,k ) × j=1 k=1

(11.127)

j=1 k=1

Lh means that the terms with j = M and N ≤ k ≤ N − 1 which are on the where k=1 path are omitted. The expression in the numerator of (11.127) is of the form of a partition function for an Ising counting lattice with the bonds zk on the path connecting the sites M, N with M, N replaced by zh−1 as illustrated in Fig. 11.16.

M +1 M

zh−1

zh

zh−1

zh−1

zh

M −1 N

N

Fig. 11.16 The Ising counting lattice with the bonds zh on the straight line path connecting sites (M, N ) and (M, N ) replaced by zh−1 .

It follows that the numerator can be written as the Pfaffian of a matrix Ac . Thus we have


σM,N σM,N = zhN −N PfAc /PfAc

¿

(11.128)

where the difference δ = Ac − Ac is given by  R 0 zh−1 − zh L  0 0 δ(0, k; 0, k + 1) = −δ T (0, k + 1; 0, k) = U 0 0 D 0 0

0 0 0 0

 0 0  0 0

(11.129)

if N ≤ k ≤ N −1 and zero otherwise. Thus, if we define y as the 2(N −N )×2(N −N ) submatrix of δ in the subspace where δ does not vanish identically and if we define Q as the 2(N − N ) × 2(N − N ) submatrix of AIc−1 in this same subspace, we find 2(N −N )

σM,N σM,N 2 = zh

2(N −N )

= zh

2(N −N )

det(AIc + δ)/detAIc = zh

det(1 + AIc−1 δ)

dety det(y −1 + Q).

(11.130)

Explicitly we see from (11.129) MN R MN + 1 R .. .. . . y=

MN − 1 R MN + 1 L MN + 2 L .. .. . . MN

L

 0 y0 0 . . . 0  0 0 y0 · · · 0     .. .. ..  .. ..  . . . . .    0 0 ··· 0 0 0 · · · y0     −y0 0 ... 0 0 0 ··· 0    0 −y0 · · · 0 0 0 ··· 0    .. .. .. .. .. .. .. ..   . . . . . . . . 0 0 · · · −y0 0 0 · · · 0 

0 0 .. .

0 ··· 0 ··· .. .. . .

(11.131)

with y0 = zh−1 − zh . Therefore dety = (zh−1 − zh )2(N −N

)

and y −1 is trivially computed. Therefore we obtain σM,N σM,N 2 = (1 − zh2 )2(N −N −1

A

0 (M, N + 1; M, N )RR .. .

A−1 (M, N − 1; M, N )RR A−1 (M, N + 1; M, N )LR × +(zh−1 − zh )−1 −1 A (M, N + 2; M, N )LR .. . A−1 (M, N ; M, N )LR

)

· · · A−1 (M, N ; M ; N − 1)RR · · · A−1 (M, N + 1, M, N − 1)RR .. .. . . ··· 0 · · · A−1 (M, N + 1; M, N − 1)LR · · · A−1 (M, N + 2; M, N − 1)LR .. .. . . ···

A−1 (M, N ; M, N − 1)LR +(zh−1 − zh )−1

(11.132)

¿


A−1 (M, N ; M, N + 1)RL −(zh−1 − zh )−1 −1 A (M, N + 1; M, N + 1)RL .. .

···

A−1 (M, N ; M, N )RL

· · · A−1 (M, N + 1; M, N )RL .. .. . .

A−1 (M, N − 1; M, N + 1)RL · · · A−1 (M, N − 1; M, N )RL −(zh−1 − zh )−1 −1 0 · · · A (M, N + 1; M, N )LL −1 A (M, N + 2; M, N + 1)LL · · · A−1 (M, N + 2; M, N )LL .. .. .. . . . A−1 (M, N ; M, N + 1)LL

···

(11.133)

0

where A−1 ≡ AIc−1 . It remains to compute the inverse matrix elements needed in (11.133). For arbitrary M, N and N this is done in [21] and the results have been used to compute the boundary correlations presented in chapter 10. We are most interested here in the case (say M = [Lv /2]) where in the thermodynamic limit the row M is infinitely far from the boundaries of the cylinder. In this limit the matrix elements of AIc−1 needed −1 for (11.133) are identical to the corresponding matrix elements of AIP ++ . This matrix is cyclic in both the vertical and horizontal directions and it is easily verified that the inverse matrix elements (in the thermodynamic limit) are AIP −1 (j , k, ; j, k)=

1 2π

2π

dθh 0

1 2π

2π

dθv eiθh (k −k)+iθv (j

−j)

AIP −1 (θh .θv ) (11.134)

0

where the 4 × 4 matrix AIP (θh , θv ) is given by (11.114). The inverse of AIP (θh , θv ) is readily computed as 1 × AIP −1 (θh , θv ) = ∆(θh , θv )   2izv sin θv R b + b∗ − abb∗ 2 − ab∗ 2 − ab ∗ ∗  L  −2 + a∗ b∗ 2 − a∗ b  −b ∗ −b + ∗a b b −2izv sin θv  U  −2 + a b 2 − ab −2izh sin θh a + a∗ − aa∗ b  D −2 + a∗ b∗ −2 + ab∗ −a∗ − a + a∗ ab∗ 2izh sin θh (11.135) where a = 1 + zh eiθh b = 1 + zv eiθv b + b∗ − abb∗ = 1 − zv2 − zh (1 + zv2 + 2zv cos θh )

(11.136)

and ∆(θh , θv ) = (1 + zh2 )(1 + zv2 ) − 2zh (1 − zv2 ) cos θh − 2zv (1 − zh2 ) cos θv . Thus we see that, in (11.133),

(11.137)


A−1 (M, N ; M, N )RR = A−1 (M, N ; M, N )LL = 0

¿

(11.138)

so that σM,N σM,N reduces to an (N − N ) × (N − N ) determinant. Thus, using the RL element of (11.135), carrying out the integral over θv , simplifying the result and noting that because of translational invariance the correlation depends only on N − N we obtain the final result (10.45)–(10.47) that in the thermodynamic limit in the interior of the lattice a0 a1 σ0,0 σ0,N = . ..

a−1 a0 .. .

· · · a−N +1 · · · a−N +2 .. .

(11.139)

aN −1 aN −2 · · · a0 where 1 an = 2π where

C(eiθ ) =

2π

dθe−inθ C(eiθ )

(11.140)

0

(1 − α1 eiθ )(1 − α2 e−iθ ) (1 − α1 e−iθ )(1 − α2 eiθ )

1/2 (11.141)

with v 1 − zv = e−2E β tanh E h β 1 + zv v 1 − zv α2 = zh−1 = e−2E β coth E h β 1 + zv

α1 = zh

(11.142)

and the square roots are defined to be positive at θ = π. 11.3.2

The diagonal correlation σ0,0 σN,N

The result (11.139) which expresses σ0,0 σ0,N as an N × N determinant is by no means unique since even the size of the determinant depends on the choice of the path joining the two spins. This arbitrariness in the choice of path may often be exploited to find useful forms of determinental representations which are difficult to see by merely looking at the determinants themselves. As an example of this phenomenon consider the correlation on the diagonal of the square lattice with periodic boundary conditions σ0,0 σN,N . If the path connecting the two spins is chosen to be the stairstep path which goes from (0, 0) to (N, N ) in 2N steps then the construction of the previous subsection gives a determinant of size 4N × 4N. However, a representation in terms of a determinant of size N × N may be obtained if we first consider a triangular lattice with diagonal bonds of strength Ed between (j, k) and (j + 1, k + 1). This problem may be reduced to a dimer problem on a lattice obtained from the triangular Ising lattice by replacing each Ising site by a cluster of six dimer sites [14]. On this triangular lattice we then consider a path of length N which goes from (0, 0) to (N, N ) along the diagonal bonds. Using this representation the correlation is computed as a 2N × 2N determinant. The result for the square lattice is then regained by letting E d → 0. In this limit the 2N × 2N

¿


determinant for σ0,0 σN,N 2 reduces to the square of a N × N determinant and the simple result (10.47) is obtained that the diagonal correlation σ0,0 σN,N is given by the Toeplitz determinant (11.139) with α1 = 0,

α2 = (sinh 2E h β sinh 2E v β)−1 .

(11.143)

Full details are given in [14, pp186–199] 11.3.3

Correlations near the boundary

As the final example of the arbitrariness in the choice of path in the computation of correlation functions consider σM,N σM,N on row M which is a fixed distance from the boundary of the cylindrical lattice which has a magnetic field interacting with the spins in row 1. Again, if a straight line path is drawn from (M, N ) to (M, N ) the correlation is computed as a 2(N − N ) × 2(N − N ) determinant. However, on the expanded lattice of Fig. 11.15 where an extra row (called zero) of spins which interact with themselves with infinite strength has been added to incorporate the magnetic field we may draw an alternative path consisting of a vertical line from (M, N ) to (0, N ), a horizontal line from (0, N ) to (0, N ) and a vertical line from (0, N ) to (M, N ). On the segment of this path from (0, N ) to (0, N ) the replacement of the weight x → x−1 results in no change at all because x = 1. Therefore the size of the determinant instead of being 2(2M + N − N ) × 2(2M + N − N ) is only 4M × 4M which has the extremely useful property that it is independent of the separation of the spins N − N . This is the reason why correlation functions at a finite distance from the boundary of the cylindrical lattice may be studied in an elementary manner. This representation was first used in [21] to obtain the results presented in chapter 10.

References [1] L. Onsager, Crystal statistics I. A two dimensional model with an order-disorder transition, Phys. Rev. 65 (1944) 117–149. [2] B. Kaufman, Crystal statistics II. Partition function evaluated by spinor analysis, Phys. Rev. 76 (1949) 1232–1243. [3] B. Kaufman and L. Onsager, Crystal statistics III. Short-range order in a binary Ising lattice, Phys. Rev. 76 (1949) 1244–1252. [4] T.D. Schultz, D.C. Mattis and E.H. Lieb, Two-dimensional Ising model as a soluble problem in many fermions, Rev. Mod. Phys. 36 (1964) 856–871. [5] M. Kac and J.C. Ward, A combinatorial solution of the two-dimensional Ising model, Phys. Rev. 88 (1952) 1332–1337. [6] R.B. Potts and J.C. Ward, The combinatorial method and the two dimensional Ising model, Prog. Theo. Phys. (Kyoto) 13 (1955) 38–46. [7] C.A. Hurst and H.S. Green, New solution of the Ising problem for a rectangular lattice, J.Chem.Phys. 33 (1960) 1059. [8] P.W. Kasteleyn, The statistics of dimers on a lattice 1. The number of dimer arrangements on a quadratic lattice, Physica 27 (1961) 1209–1225. [9] P.W. Kasteleyn, Dimer statistics and phase transitions, J.Math. Phys. 4 (1963) 287–293. [10] P.W. Kasteleyn in Graph theory and theoretical physics, ed. F. Harary,(Academic Press, New York 1967) 43. [11] R.J. Baxter and I.G.Enting, 399th solution of the Ising model, J. Phys. A11 (1978) 2463–2473. [12] R.J. Baxter, Exactly solved models in Statistical Mechanics, Academic Press, London (1982). [13] G.H. Wannier, The statistical problem in cooperative phenomena, Rev. Mod. Phys.17 (1945) 50–60. [14] B.M. McCoy and T.T. Wu, The two dimensional Ising model (Harvard University Press 1973). [15] M.E. Fisher, On the dimer solution of planar Ising models, J.Math. Phys. 7 (1966) 1776–1781. [16] G. Tesler, Matchings in graphs on non-orientable surfaces, J. Comb. Theo. B 78 (2002) 198–231. [17] R. Costa-Santos and B.M. McCoy, Dimers and the critical Ising model on lattices of genus > 1, Nucl. Phys. B623 [FS]( 2002) 439–473. [18] A.E. Ferdinand, Statistical mechanics of dimers on a quadratic lattice, J.Math. Phys. 8 (1967) 2332–2339. [19] P. Fendley, R. Moessner and S.L. Sondhi, Classical dimers on the triangular lattice, Phys. Rev. B66 (2002) 214513-(1–14).

¿

References

[20] W.T. Lu and F.Y. Wu, Close-packed dimers on nonorientable surfaces, Phys. Lett. A 293 (2002) 235–246. [21] B.M. McCoy and T.T. Wu, Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. IV, Phys. Rev. 162 (1967) 436–475. [22] E.W. Montroll, R.B. Potts and J.C. Ward, Correlations and spontaneous magnetization of the two-dimensional Ising model, J.Math. Phys. 4 (1963) 308–322.

12 Ising model spontaneous magnetization and form factors In the preceding chapter we demonstrated that every correlation function σ0,0 σM,N of the two-dimensional Ising model in zero external field can be expressed a determinant (in an infinite number of ways). The simplest such results are for the correlation σ0,0 σN,N on the diagonal and σ0,0 σ0,N in a row which were both shown to be given as an N × N Toeplitz determinant c0 c1 = c2 .. .

DN

c−1 c0 c1 .. .

c−2 c−1 c0 .. .

· · · c−N +1 · · · c−N +2 · · · c−N +3 .. .. . .

(12.1)

cN −1 cN −2 cN −3 · · · c0 with 1 cn = 2π where

2π


(12.2)

0

(1 − α1 eiθ )(1 − α2 e−iθ ) C(e ) = (1 − α1 e−iθ )(1 − α2 eiθ )

1/2

iθ

(12.3)

and the square root is defined to be positive at θ = π. For the diagonal correlation α1 = 0,

α2 = (sinh 2E v β sinh 2E h β)−1

(12.4)

and for the row correlation α1 = e−2E

v

β

tanh E h β,

α2 = e−2E

v

β

coth E h β.

(12.5)

These determinants give a very efficient method of computing the correlations when the separation N is small. Examples of these results were given in chapter 10. However, the direct expansion of these determinants is not an efficient way to obtain the behavior of the correlations when the separation N is large. In chapter 10 we gave alternative expressions for the correlations in terms of the form factor expansion which are useful in the large N regime. In this chapter we will prove these form factor expansions for σ0,0 σN,N and σ0,0 σ0,N by the use of

¿

Ising model spontaneous magnetization and form factors

Wiener–Hopf sum equations. The solution of Wiener–Hopf sum equations is presented in section 12.1. The leading term in the form factor expansion for T < Tc is the spontaneous magnetization. This is computed in section 12.2 by proving first a much more general theorem due to Szegö. In section 12.3 we compute the form factor expansions both above and below Tc . In section 12.4 we compute the leading behavior as N → ∞ for the three cases of T > Tc , T = Tc and T < Tc . We conclude in section 12.5 by (n) presenting the method whereby the diagonal form factor fN (t) which is given as an n dimensional integral can be reduced to the sum of products of the hypergeometric functions F (−1/2, 1/2; 1; t) and F (1/2, 1/2; 1; t) where t = α22 for T < Tc and α−2 2 for T > Tc which were given in chapter 10.

12.1

Wiener–Hopf sum equations

A Wiener–Hopf sum equation is the set of simultaneous linear equations ∞

cn−m xm = yn

0 ≤ n.

(12.6)

m=0

These sum equations are distinguished from the most general set of linear equations in two respects: i) The numbers cm−n depend only on the difference m − n and not on m and n separately and ii) the upper limit of of summation runs to infinity. Because of condition ii) we need some conditions on cn , xn and yn to make the problem well defined and for our applications it is sufficient to impose the conditions ∞

|yn | < ∞

n=−∞ ∞

|cn | < ∞

n=−∞ ∞

|xn | < ∞.

(12.7)

n=−∞

We will solve these equations under the further condition that ln C(eiθ ), with C(e−iθ ) defined by (12.11), is continuous, periodic and zero free for 0 ≤ θ ≤ 2π. We follow the procedure in [1, chapter 9]. 12.1.1

Fourier transforms

We begin by considering an equation simpler than (12.6) where the lower limit of summation extends to minus infinity ∞

cn−m xm = yn

− ∞ < n < ∞.

m=−∞

To solve (12.8) we define the three Fourier transforms on the unit circle |ξ| = 1

(12.8)

Wiener–Hopf sum equations ∞

X(ξ) = Y (ξ) = C(ξ) =

n=−∞ ∞ n=−∞ ∞

¿

xn ξ n

(12.9)

yn ξ n

(12.10)

cn ξ n

(12.11)

n=−∞

which, because of (12.7), exist and are continuous on |ξ| = 1. Multiply (12.8) by ξ n and sum n from −∞ to ∞. Using (12.7) we may interchange the order of the double sums to find Y (ξ) =

∞ n=−∞

ξn

∞

cn−m xm =

m=−∞

∞

xm ξ m

m=−∞

∞

cn−m ξ n−m = C(ξ)X(ξ).

n=−∞

(12.12) Then if we make the further restriction that C(ξ) = 0 for |ξ| = 1 we may solve (12.12) as X(ξ) = Y (ξ)/C(ξ). (12.13) and thus we may invert (12.9) to find the solution Y (ξ) 1 xn = . dξξ −n−1 2πi |ξ|=1 C(ξ)

(12.14)

We now consider applying this same procedure of Fourier transform to the original Wiener–Hopf sum equation (12.6). Now xn , yn and the equations (12.6) exist only for n ≥ 0. In order to execute a Fourier transform we thus need to make the further definitions xn = 0 for n ≤ −1 yn = 0 for n ≤ −1 ∞ cn−m xm for n ≤ −1 vn =

(12.15) (12.16) (12.17)

m=0

=

n≥0

0 for

(12.18)

and thus we may write (12.6) as ∞

cn−m xm = yn + vn

for − ∞ < n < ∞.

(12.19)

m−∞

We may now define Fourier transforms on |ξ| = 1 by (12.11) and X(ξ) =

∞ n=0

xn ξ n

(12.20)

¿


Y (ξ) =

∞

n=0 −1

V (ξ) =

yn ξ n

(12.21)

vn ξ n

(12.22)

n=−∞

and multiply (12.19) by ξ n and sum n from −∞ to ∞ to obtain C(ξ)X(ξ) = Y (ξ) + V (ξ).

(12.23)

Equation (12.23) differs from (12.12) in that it contains two unknown functions X(ξ) and V (ξ) instead of only one unknown. Thus further information is needed to solve the problem. 12.1.2

Splitting and factorization

Consider any function F (ξ) which has a convergent Laurent expansion on the unit circle |ξ| = 1 ∞ F (ξ) = fn ξ n (12.24) n=−∞

with

∞

|fn | < ∞.

(12.25)

n=−∞

Then we may define what we call a + function [F (ξ)]+ which is continuous on |ξ| = 1, and analytic for |ξ| < 1 by ∞ fn ξ n (12.26) [F (ξ)]+ = n=0

and we can define a − function [F (ξ)]− continuous on |ξ| = 1, analytic for |ξ| > 1 and vanishing as ξ → ∞ as −1 [F (ξ)]− = fn ξ n . (12.27) n=−∞

Clearly for |ξ| = 1 F (ξ) = [F (ξ)]+ + [F (ξ)]− . Furthermore we have from Cauchy’s theorem for |ξ| > 1 [F (ξ )]− 1 [F (ξ)]− = − dξ 2πi |ξ |=1 ξ −ξ and 0=−

1 2πi

|ξ |=1

dξ

[F (ξ )]+ ξ − ξ

(12.28)

(12.29)

(12.30)

where the integration contour is counterclockwise on |ξ | = 1 and thus using (12.28) we find that, for |ξ| > 1,

Wiener–Hopf sum equations

[F (ξ)]− = −

1 2πi

Similarly, for |ξ| < 1 [F (ξ)]+ =

1 2πi

dξ |ξ |=1

|ξ |=1

dξ

F (ξ ) . ξ − ξ

F (ξ ) . ξ − ξ

¿

(12.31)

(12.32)

The decomposition (12.28) is called a(n) (additive) splitting of the function F (ξ). We now consider making a factorization of the function C(ξ) on |ξ| = 1 as C(ξ) = P −1 (ξ)Q−1 (ξ −1 )

(12.33)

where both P (ξ) and Q(ξ) are both + functions (continuous for |ξ| = 1 and analytic for |ξ| < 1) which are nonzero for |ξ| < 1. When C(ξ) has the property that ln C(ξ) is continuous and periodic on |ξ| = 1 we may obtain the factorization (12.33) by first taking the logarithm ln C(ξ) = − ln P (ξ) − ln Q(ξ −1 ) (12.34) where ln P (ξ) is a + function and ln Q(ξ −1 ) is a − function. Thus by making an additive splitting of ln C(ξ) ln C(ξ) = [ln C(ξ)]+ + [ln C(ξ)]− .

(12.35)

Thus by comparing (12.34) and (12.35) we find that P (ξ) = exp(−[ln C(ξ)]+ )

(12.36)

Q(ξ −1 ) = exp(−[ln C(ξ)]− ).

(12.37)

which satisfies the requirement that P (ξ) and Q(ξ) are + functions which are nonzero for |ξ| < 1. It follows from (12.26) and (12.37) that [Q(ξ −1 )]+ = Q(0) = 1. 12.1.3

(12.38)

Solution

We may now use the factorization (12.33) in (12.23) to write P −1 (ξ)Q−1 (ξ −1 )X(ξ) = Y (ξ) + V (ξ)

(12.39)

and multiply by Q(ξ −1 ) to find P −1 (ξ)X(ξ) = Q(ξ −1 )Y (ξ) + Q(ξ −1 )V (ξ).

(12.40)

We now recognize that P −1 (ξ)X(ξ) is a + function (because both P −1 (ξ) and X(ξ) are + functions) and that Q(ξ −1 )V (ξ) is a − function (because Q(ξ) is a + function and V (ξ) is a − function). However, Q(ξ −1 )Y (ξ) contains both + and − parts. Thus we make the additive splitting of Q(ξ −1 )Y (ξ) Q(ξ −1 )Y (ξ) = [Q(ξ −1 )Y (ξ)]+ + [Q(ξ −1 )Y (ξ)]−

(12.41)

and write (12.40) for |ξ| = 1 P −1 (ξ)X(ξ) − [Q(ξ −1 )Y (ξ)]+ = [Q(ξ −1 )Y (ξ)]− + Q(ξ −1 )V (ξ).

(12.42)

The left-hand side defines a function analytic for |ξ| < 1 and continuous on |ξ| = 1 and the right-hand side defines a function which is analytic for |ξ| > 1 and is continuous

¿


for |ξ| = 1. Taken together they define a function E(ξ) analytic for all ξ except possibly for |ξ| = 1 and continuous everywhere. But these properties are sufficient to prove [1, pp.211-213] that E(ξ) is an entire function which vanishes at |ξ| = ∞ and thus, by Liouville’s theorem, must be zero everywhere. Therefore both the right-hand side and the left-hand side of (12.42) vanish separately and thus we have

and

X(ξ) = P (ξ)[Q(ξ −1 )Y (ξ)]+

(12.43)

V (ξ) = −Q−1 (ξ −1 )[Q(ξ −1 )Y (ξ)]− .

(12.44)

Thus inverting the Fourier transform we have the desired solution 1 1 dξ n+1 P (ξ)[Q(ξ −1 )Y (ξ)]+ . xn = 2πi |ξ|=1 ξ

12.2

(12.45)

Spontaneous magnetization and Szeg¨ o’s theorem

The spontaneous magnetization M (T )− is obtained from the two-point function as lim

σ0,0 σM,N = M (T )2− .

M 2 +N 2 →∞

(12.46)

In particular because this limit should be independent of how M and N go to infinity it should be sufficient to compute either lim σ0,0 σN,N

(12.47)

lim σ0,0 σ0,N .

(12.48)

N →∞

or N →∞

Both of these correlations are given in (12.1) as N × N Toeplitz determinants. We thus need to compute the value of this determinant when its size becomes infinite. The mathematics needed to compute this limit was specifically invented to study this Ising model problem. However, once the technique was discovered it was found to lead to results of much greater generality. We will therfore compute the spontaneous magnetization of the Ising model by a very general theorem found by Szegö [2]. Szeg¨ o’s theorem. If the generating function C(ξ) and lnC(ξ) are continuous on the unit circle |ξ| = 1 then the behavior for large N of the Toeplitz determinant

DN

c0 c1 = c2 .. .

c−1 c0 c1 .. .

c−2 c−1 c0 .. .

· · · c−N +1 · · · c−N +2 · · · c−N +3 .. .. . .

(12.49)

cN −1 cN −2 cN −3 · · · c0 with cn =

1 2π

0

2π


(12.50)


is given by

N

lim DN /µ

ng−n gn

(12.51)

dθ ln C(e )

(12.52)

dθe−inθ ln C(eiθ )

(12.53)

= exp

N →∞

∞

¿

n=1

where

1 µ = exp 2π and gn =

1 2π

2π

2π

iθ

0

0

wherever the sum in (12.51) converges. 12.2.1

Proof of Szeg¨ o’s theorem

The proof of Szeg¨ o’s theorem has a long history which is given in [3]. The result actually proven by Szegö [2] required that the generating function C(eiθ ) be real and positive which is not satisfied in the Ising case. We follow the presentation in [1, chapter 10] and separate the derivation of the result (12.51) into two parts by first computing the ratio DN /DN +1 as N → ∞ to find µ of (12.52) and then computing the limit (12.51). Behavior of DN /DN +1 for N → ∞ We define µN as the ratio µN = DN +1 /DN (N )

and define the quantities xn N

(12.54)

as the solution of the set of N + 1 linear equations

(N ) cn−m xm = δn,0 for 0 ≤ n ≤ N.

(12.55)

m=0 (N )

These equations have a unique solution for xn

if

DN +1 = 0

(12.56)

and thus by Cramer’s rule µN = DN +1 /DN = [x0 ]−1 . (N )

(12.57)

Therefore, assuming that the limit (∞)

x0

(N )

= lim x0 N →∞

(12.58)

exists, this limit may be computed by solving the Wiener–Hopf sum equation for n ≥ 0 ∞

(N ) cn−m xm = δn,0 .

m=0

The solution of (12.59) is given by (12.45) with

(12.59)

¿


Y (ξ) =

∞

ξ n δn,0 = 1

(12.60)

1 P (ξ)[Q(ξ −1 )]+ . ξ n+1

(12.61)

n=0

and thus we obtain x(∞) n

1 = 2πi

dξ |ξ|=1

Thus, using (12.38) for [Q(ξ −1 )]+ , we obtain 1 1 (∞) xn = dξ P (ξ). 2πi |ξ|=1 ξ n+1

(12.62)

Therefore if we set n = 0 and use the fact that P (ξ) is analytic for |ξ| ≤ 1 we find that (∞)

x0

= P (0)

(12.63)

from which, using the computation (12.36) for P (ξ) in terms of C(ξ) we find that when ln C(ξ) is continuous on |ξ| = 1 that 1 dξ (∞) µ−1 = lim DN /DN +1 = x0 = P (0) = exp − ln C(ξ) N →∞ 2πi |ξ|=1 ξ 2π 1 iθ = exp − dθ ln C(e ) (12.64) 2π 0 which is (12.52). If we assume that there exists an N0 such that DN = 0 for N ≥ N0 then from (12.57) it follows that lim DN /µN = lim DN0 /µN0 +1

N →∞

N →∞

N −1

µn /µ.

(12.65)

n=N0

Thus the limit in (12.51) will exist and be nonzero if ∞

|1 − µ/µN | < ∞.

(12.66)

N =N0

Computation of limN →∞ DN /µN We will prove (or at least discover) (12.51) by studying the dependence of the limiting value of DN /µN as N → ∞ on the generating function C(eiθ ). For this purpose, we compare DN defined by (12.49) with the determinant

¯N D

c¯0 c¯1 = c¯2 .. .

c¯−1 c¯0 c¯1 .. .

c¯−2 c¯−1 c0 .. .

· · · c¯−N +1 · · · c¯−N +2 · · · c¯−N +3 .. .. . .

c¯N −1 c¯N −2 c¯N −3 · · · c¯0

(12.67)


with

2π

¯ iθ ) dθe−inθ C(e

(12.68)

¯ iθ ) = C(eiθ )(1 − αe−iθ ) C(e

(12.69)

|α| < 1.

(12.70)

c¯n = and

1 2π

¿

0

with From the condition (12.70) it follows that exp

1 2π

2π 0

2π ¯ iθ ) = exp 1 dθ ln C(e dθ ln C(eiθ ) = µ. 2π 0

(12.71)

It follows from (12.50), (12.68) and (12.69) that c¯n = cn − αcn+1

(12.72)

and thus from (12.67) that

¯N D

c0 − αc1 c1 − αc2 = c2 − αc3 .. .

c−1 − αc0 c0 − αc1 c1 − αc2 .. .

· · · c−N +1 − αc−N +2 · · · c−N +2 − αc−N +3 · · · c−N +3 − αc−N +4 .. .. . .

(12.73)

cN −1 − αcN cN −2 − αcN −3 · · · c0 − αc1 and this may be written as an (N + 1) × (N + 1) determinant

¯N D

1 c1 c2 = c3 .. .

α c0 c1 c2 .. .

α2 c−1 c0 c1 .. .

α3 c−2 c−1 c0 .. .

· · · αN · · · c−N +1 · · · c−N +2 · · · c−N +3 .. .. . .

(12.74)

cN cN −1 cN −2 cN −3 · · · c0 which may be verified by subtracting α times column N from column N + 1, then α times column N − 1 from column N and continuing until we subtract α times column one from column 2. (N ) We now define the N + 1 quantities x¯j with 0 ≤ j ≤ N as the solutions of the N + 1 linear equations N (N ) αm x ¯m =1 (12.75) m=0

and

N m=0

(N ) cn−m x ¯m = 0,

for 1 ≤ n ≤ N.

(12.76)

¿


Then again by Cramer’s rule (N )

¯N. = DN / D

x ¯0

(12.77)

We now let N → ∞ and assume as before that (N )

lim x¯0

N →∞

(∞)

where from (12.75) and (12.76) the x ¯j

∞

(∞)

= x¯0

(12.78)

for 0 ≤ j satisfy

αm x ¯(∞) m =1

(12.79)

m=0

and

∞

cn−m x ¯(∞) m = 0,

for 1 ≤ n.

(12.80)

m=0

Equations (12.79) and (12.80) taken together are not a set of Wiener–Hopf sum equations. However, if we define y0 by ∞

c−m x ¯(∞) m = y0

(12.81)

m=0

then (12.80) and (12.81) taken together do form a set of Wiener–Hopf sum equations and thus if we define ∞ n ¯ (∞) (ξ) = X x ¯(∞) (12.82) n ξ n=0

we find from (12.43) that ¯ (∞) (ξ) = P (ξ)[Q(ξ −1 )Y (ξ)]+ = y0 Q(0)P (ξ). X

(12.83)

In order to determine y0 we note that from (12.79) ¯ (∞) (α) = 1 X

(12.84)

y0 Q(0)P (α) = 1.

(12.85)

¯ (∞) (ξ) = P (ξ) X P (α)

(12.86)

and thus using (12.83) we find

Therefore from (12.83)

and thus, from (12.77), we obtain the result DN (∞) ¯ (∞) (0) = P (0) . lim ¯ = x ¯0 = X P (α) DN

N →∞

Finally we use the expression (12.36) for P (ξ) in the form

(12.87)


P (ξ) = P (0) exp −

∞

¿

gn ξ

n

,

(12.88)

n=1

where gn is defined by (12.53), to find that (12.87) reduces to ∞ DN gn αn . lim ¯ = exp N →∞ DN n=1

(12.89)

We now use the definition of gn of (12.53) and make the analogous definition 2π 1 ¯ iθ ) g¯n = dθe−inθ ln C(e (12.90) 2π 0 to find g¯n = gn for n ≥ 0 = gn + α−n /n for n < 0

(12.91) (12.92)

and thus we may rewrite (12.89) as ∞ DN n(g−n gn − g¯−n g¯n ). lim ¯ = exp N →∞ DN n=1

(12.93)

We next consider the same problem but instead of (12.69) we have ¯ iθ ) = C(eiθ )(1 − α C(e ¯ eiθ )

(12.94)

with |¯ α| < 1 and for this we can apply (12.93) to the comple conjugate functions to find that (12.93) applies in this case as well. We now repeat the process a finite number of times to find that for ¯ iθ ) = C(eiθ ) C(e

n1

(1 − α(n) e−iθ )

n=1

n2

(1 − α ¯ (n) eiθ )

(12.95)

n=1

α(n) | < 1 that with |α(n) | < 1 and |¯ lim

N →∞

DN ¯ (n1 ,n2 ) D N

= exp

∞

(n ,n2 ) (n1 ,n2 ) gk ).

k(g−k gk − g¯−k1

(12.96)

k=1

To obtain Szegö’s theorem we now need to consider the special limiting case ¯ iθ ) = 1. Then g¯n = 0 and we obtain from (12.96) the n1 , n2 → ∞ such that C(e desired result (12.51). To justify this we must interchange the limit N → ∞ with the limits n1 , n2 → ∞. The validity of this interchange will depend on the smoothness of the generating function C(eiθ ) on 0 ≤ θ ≤ 2π. A proof of this interchange under conditions which are sufficient for the generating function (12.3) which is analytic on 0 ≤ θ ≤ 2π is given in [1]. The most general restrictions known are fully presented in [3]. Thus Szeg¨ o’s theorem is proven.

¿


12.2.2

The spontaneous magnetization

It remains to apply Szegö’s theorem to the diagonal and row correlations given by (12.1) with C(eiθ ) given by (12.3). This function has the property that C(e−iθ ) = 1/C(eiθ )

(12.97)

and hence we find from (12.52) that µ = 1.

(12.98)

Furthermore because 0 ≤ α1 ≤ α2 < 1 we use the formula ln(1 − αj eiθ ) = −

∞

n−1 (αj eiθ )n

(12.99)

n=1

in the definition of gn (12.53) to find for n > 0 1 n [α − αn1 ] 2n 2 1 = − [αn2 − αn1 ] 2n

gn =

(12.100)

g−n

(12.101)

Thus we find ∞

ng−n gn = −

n=1

=−

∞ 1 −1 n n [α2 − αn1 ]2 4 n=1

∞ 1 −1 2n 1 (1 − α22 )(1 − α21 ) n n ln . n [α2 + α2n − 2α α ] = 1 1 2 4 n=1 4 (1 − α1 α2 )2

(12.102)

Therefore we find from Szegö’s theorem (12.51) that for the diagonal correlation where α1 and α2 are given by (12.4) M 2 = lim σ0,0 σN,N = [1 − (sinh 2E v β sinh 2E h β)−2 ]1/4

(12.103)

N →∞

Similarly when we consider the row correlation function σ0,0 σ0,N where α1 and α2 are given by (12.5) we also find that

(1 − α22 )(1 − α21 ) lim σ0,0 σ0,N = N →∞ (1 − α1 α2 )2 = [1 − (sinh 2E v β sinh 2E h β)−2 ]1/4 .

1/4

(12.104)

Thus, as expected, the limit N → ∞ of both the diagonal and the row correlation are the same, and we conclude that the spontaneous magnetization of the Ising model is M = [1 − (sinh 2E v β sinh 2E h β)−2 ]1/8 . (12.105) This result was first announced by Onsager [4] in 1949 and proven by Yang [5] in 1952.

Form factor expansions of C(N, N ) and C(0, N )

12.3

¿


The representation of the diagonal C(N, N ) and row C(0, N ) correlation functions as N × N determinants is very efficient for computation when N is small. However, this representation is not efficient when N is large, and this large N behavior is mandatory for the microscopic understanding of critical behavior presented in chapter 5. Therefore in this section we will extend the considerations which we used to derive Szegö’s theorem to recast the row and diagonal correlations into the form factor forms of (10.76)–(10.79) from which the large N limit for both T < Tc and T > Tc can be easily determined. The calculations are different for T < Tc and T > Tc and will be treated separately. We follow the treatment of [6] which extends the Wiener–Hopf methods of [7]. 12.3.1

Expansion for T < Tc

The row and diagonal correlations are given in (12.1) by a determinant DN of the form (12.49) with C(eiθ ) given by (12.3) and for T < Tc we have seen that ln C(ξ) is continuous on |ξ| = 1. We begin the derivation of the form factor expansions of chapter 10 by writing ∞ Dn 2 DN = M (12.106) Dn+1 n=N

where

M 2 = (1 − t)1/4 with t = (sinh 2E h β sinh 2E v β)−2

(12.107)

is the limiting value as N → ∞ previously computed from Szeg¨ o’s theorem. We will proceed in three steps. First we extend the calculation of the previous section to obtain an exact expression for the ratio DN /DN +1 valid for all N . We will then show how the product in (12.106) can be put in the exponential form of (10.67)–(10.69). Finally we will expand the exponential to obtain the form factor expression (10.76). The ratio DN /DN +1 The ratio DN /DN +1 has already been seen in the previous section to be given by (N )

x0 (N )

where the xn

= DN /DN +1

(12.108)

are determined from the set of linear equations N

(N ) cn−m xm = δn,0

for 0 ≤ n ≤ N.

(12.109)

m=0

We will prove in this subsection that (N )

DN /DN +1 = x0

=

∞ n=0

(0)

(2k)

where φN = 1 and for k ≥ 1 φN

is defined as

(2n)

φN

(12.110)

¿


(2k)

φN

= (−1)k+1

1 (2π)2k

2k

dzj zjN +1

j=1

k

−1 −1 {Q(z2j−1 )Q(z2j−1 )P (z2j )P (z2j )}

j=1 −1 × z1−1 z2k

2k−1 l=1

1 1 − zl zl+1

(12.111)

where the contours are such that |zi | = 1 − , and the functions P (ξ) and Q(ξ) are the functions in the Wiener–Hopf splitting (12.33) of C(eiθ ) (12.3) which for our problem with T < Tc are explicitly given by 1/2 1 − α1 ξ = 1/P (ξ). (12.112) Q(ξ) = 1 − α2 ξ We follow the procedure of [7] and generalize the Wiener–Hopf procedure by defining xn(N ) = yn = 0, for n ≤ −1 and n ≥ N + 1 yn =

N

(N ) cn−m xm for 0 ≤ n ≤ N

(12.113) (12.114)

m=0

vn(N )

=

N

(N ) c−n−m xm for n ≥ 1

m=0

= 0 for n ≤ 0 un(N ) =

N

(12.115)

(N ) cN +n−m xm for n ≥ 1

m=0

= 0 for n ≤ 0.

(12.116)

We further define XN (ξ) =

N

xn(N ) ξ n

(12.117)

n=0

Y (ξ) =

N

yn ξ n

n=0 ∞

UN (ξ) = VN (ξ) =

n=1 ∞

(12.118)

un(N ) ξ n

(12.119)

vn(N ) ξ n

(12.120)

n=1

and we note from (12.106) and (12.117) that (N )

DN /DN +1 = x0

= XN (0).

(12.121)


¿

We multiply (12.109) by ξ n and sum on n from −∞ to ∞ to obtain C(ξ)XN (ξ) = Y (ξ) + UN (ξ)ξ N + VN (ξ −1 )

(12.122)

and then use the factorization (12.33) in (12.122) and decompose into + and − parts to find for |ξ| = 1 that P −1 (ξ)XN (ξ) − [Q(ξ −1 )Y (ξ)]+ − [Q(ξ −1 )UN (ξ)ξ N ]+ = Q(ξ −1 )VN (ξ −1 ) + [Q(ξ −1 )Y (ξ)]− + [Q(ξ −1 )UN (ξ)ξ N ]− .

(12.123)

Then we make the identical argument made in section 12.1 to show that each side of this equation vanishes separately and thus we have the two equations XN (ξ) = P (ξ){[Q(ξ −1 )Y (ξ)]+ + [Q(ξ −1 )UN (ξ)ξ N ]+ } VN (ξ −1 ) = −Q−1 (ξ −1 ){[Q(ξ −1 )Y (ξ)]− + [Q(ξ −1 )UN (ξ)ξ N ]− }.

(12.124) (12.125)

Furthermore a second set of equations is found by considering ξ N XN (ξ −1 ) in place of XN (ξ) and thus we also find XN (ξ −1 )ξ N = Q(ξ){[P (ξ −1 )Y (ξ −1 )ξ N ]+ + [P (ξ −1 )VN (ξ)ξ N ]+ }

(12.126)

UN (ξ −1 ) = −P −1 (ξ −1 ){[P (ξ −1 )Y (ξ −1 )ξ N ]− + [P (ξ −1 )VN (ξ)ξ N ]− }.(12.127) For any function F (ξ) we will need, in addition to the functions [F (ξ)]+ and [F (ξ)]− defined in (12.26) and (12.27), the function [F (ξ)]+ =

∞

fn ξ n .

(12.128)

[F (ξ −1 )]− = [F (ξ)]+

(12.129)

n=1

From (12.24) and (12.27) we have

and [F (ξ)]+ has the integral representation [F (ξ)]+ = [F (ξ)]+ −

1 2πi

dξ

F (ξ ) 1 ξ = ξ 2πi

|ξ |=1

dξ

F (ξ ) − ξ)

ξ (ξ

(12.130)

where the contour of integration is indented outward at ξ = ξ. Thus noting that [Q(ξ −1 )]+ = 1 and Y (ξ) = 1, and using (12.112) and (12.129) we rewrite equations (12.124),(12.125) and (12.127) as XN (ξ) = P (ξ){1 + [Q(ξ −1 )UN (ξ)ξ N ]+ }

(12.131)

VN (ξ −1 ) = −P (ξ −1 ){[Q(ξ −1 )]− + [Q(ξ −1 )UN (ξ)ξ N ]− }

(12.132)

¿


UN (ξ) = −Q(ξ){[P (ξ)ξ −N ]+ + [P (ξ)VN (ξ −1 )ξ −N ]+ }.

(12.133)

When N → ∞ we see from the definition (12.116) that UN (ξ) vanishes. Therefore (1) we solve the equations (12.131)–(12.133) iteratively by defining VN (ξ) as the first approximation to VN (ξ) obtained by replacing UN (ξ) by zero in (12.132). Thus VN (ξ −1 ) = −P (ξ −1 )[Q(ξ −1 )]− . (1)

(12.134)

Furthermore because Q(ξ −1 ) is analytic for |ξ| > 1 and because Q(0) = 1 we have [Q(ξ −1 )]− = Q(ξ −1 ) − Q(0) = Q(ξ −1 ) − 1

(12.135)

therefore it follows from (12.112) and (12.135) that (12.134) becomes VN (ξ −1 ) = −P (ξ −1 )[Q(ξ −1 )]− = P (ξ −1 ) − 1. (1)

(12.136)

We define UN (ξ) by replacing VN (ξ −1 ) in (12.133) by VN (ξ −1 ) as given by equation (12.136). Thus we find (1)

(1)

UN (ξ) = −Q(ξ)[P (ξ −1 )P (ξ)ξ −N ]+ . (1)

(12.137) (1)

It thus follows from equation (12.131) that the first approximation XN (ξ) to XN (ξ) is given by XN (ξ) = P (ξ){1 − [Q(ξ −1 )Q(ξ)[P (ξ −1 )P (ξ)ξ −N ]+ ξ N ]+ } ξ N 1 dξ Q(ξ −1 )Q(ξ )[P (ξ −1 )P (ξ )ξ −N ]+ }. = P (ξ){1 − 2πi ξ −ξ (12.138) (1)

(2)

Letting ξ = 0 in equation (12.138), and using P (0) = 1, and writing X (1) (0) = 1+φN , we obtain 1 (2) φN = − dξ Q(ξ −1 )Q(ξ)[P (ξ −1 )P (ξ)ξ −N ]+ ξ N −1 2πi 1 1 1 1 N −1 dξ1 Q(ξ1 )Q(ξ1 ) ξ dξ2 =− P (ξ2−1 )P (ξ2 )ξ2−N 2πi 2πi 1 ξ2 ξ2 − ξ1 (12.139) where ξ2 is indented outward at ξ2 = ξ1 . Thus, if we set ξ2k+1 = z2k+1 (2)

−1 ξ2k = z2k

(12.140)

we obtain φN of (12.111). (2) We now continue by calculating VN (ξ −1 ), the second approximation to VN (ξ −1 ), by using (12.137) in (12.132) to find


¿

VN (ξ −1 ) = −P (ξ −1 ){[Q(ξ −1 )]− + [Q(ξ −1 )UN (ξ)ξ N ]} = −P (ξ −1 )[Q(ξ −1 )]− + P (ξ −1 )[Q(ξ −1 )Q(ξ)ξ N [P (ξ −1 )P (ξ)ξ −N ]+ ]− .(12.141) (2)

(1)

(2)

Next, we calculate UN (ξ) by using (12.141) in (12.133) with (12.135) to obtain UN (ξ) = −(P (ξ))−1 {[P (ξ)ξ −N ]+ + [P (ξ)VN (ξ −1 )ξ −N ]+ } (2)

(2)

= −Q(ξ)[P (ξ)P (ξ −1 )ξ −N ]+ −Q(ξ)[P (ξ)P (ξ −1 )ξ −N [Q(ξ)Q(ξ −1 )ξ N [P (ξ)P (ξ −1 )ξ −N ]+ ]− ]+ .

(12.142)

(2)

We will now calculate XN (ξ) from (12.131) as XN (ξ) = P (ξ){1 + [Q(ξ −1 )UN (ξ)ξ N ]+ } (2)

(2)

= P (ξ) − P (ξ)[Q(ξ −1 )Q(ξ)ξ N [P (ξ)P (ξ −1 )ξ −N ]+ ]+ −P (ξ)[Q(ξ −1 )Q(ξ)ξ N [P (ξ)P (ξ −1 )ξ −N [Q(ξ)Q(ξ −1 )ξ N [P (ξ)P (ξ −1 )ξ −N ]+ ]− ]+ ]+ . (12.143) (2)

(2)

(4)

Letting ξ = 0 in equation (12.143), we obtain XN (0) = 1 + φN + φN where 1 (4) dξ Q(ξ −1 )Q(ξ) φN = − 2πi × [P (ξ −1 )P (ξ)ξ −N [Q(ξ −1 )Q(ξ)ξ N [P (ξ −1 )P (ξ)ξ −N ]+ ]− ]+ ξ N −1 1 1 −1 N dξ1 ξ1 Q(ξ1 )Q(ξ1 ) dξ2 =− ξ −N −1 P (ξ2−1 )P (ξ2 ) (2πi)4 ξ2 − ξ1 2 1 1 dξ3 ξ3N +1 Q(ξ3−1 )Q(ξ3 ) dξ4 ξ −N −1 P (ξ4−1 )P (ξ4 )(12.144) ξ3 − ξ2 ξ4 − ξ3 4 (4)

Using the change of variables (12.140) we obtain φN of (12.111). In general, we iteratively define (from equation 12.132) (n+1)

VN

(ξ −1 ) = −P (ξ −1 ){[Q(ξ −1 )]− + [Q(ξ −1 )UN (ξ)ξ N ]− }. (n)

(12.145)

It then follows from equation (12.133) that (n)

(n−1)

UN (ξ) − UN (ξ) −1 = −Q(ξ )[P (ξ)P (ξ −1 )ξ −N [Q(ξ)Q(ξ −1 )ξ N [P (ξ)P (ξ −1 )ξ −N [Q(ξ)Q(ξ −1 )ξ N ...]− ]+ ]− ]+ (12.146) (2k)

where there are 2n − 1 brackets. It now follows from equation (12.131) that φN

is

¿


1 dξ ξ N −1 Q(ξ)Q(ξ −1 )[P (ξ)P (ξ −1 )ξ −N 2πi [Q(ξ)Q(ξ −1 )ξ N [P (ξ)P (ξ −1 )ξ −N [Q(ξ)Q(ξ −1 )ξ N ...]− ]+ ]− ]+ (2k)

φN

=−

(12.147)

where there are 2k − 1 brackets. By use of (12.140), we obtain equation (12.111) and thus from (12.106) and (12.108) we have shown that ∞ ∞

DN = (1 − t)1/4

φ(2k) m .

(12.148)

m=N k=0

Exponentiation Our next step is to show that (12.148) can be put in the form of (10.67)–(10.69). DN = (1 − t)1/4 eFN where FN =

∞

(2n)

FN

(12.149)

(12.150)

n=1

with (2n)

FN

=

(−1)n+1 1 n (2π)2n

2n j=1

n dzj zjN −1 −1 P (z2j )P (z2j )Q(z2j−1 )Q(z2j−1 ). 1 − zj zj+1 j=1

(12.151) and z2n+1 ≡z1 . The path of integration is along the unit circle |zj | = 1 − . We begin this demonstration by defining a function (2n) FÑ

(−1)n+1 1 n (2π)2n

=

(2n)

= FN

2n

n 2n ' ( dzj zjN −1 −1 Q(z2j−1 )Q(z2j−1 )P (z2j )P (z2j ) (1 − zj ) 1 − zj zj+1 j=1 j=1 j=1

(2n)

− FN +1

(12.152)

which has the property that (2n)

FN

=

∞

(2n) F˜k .

(12.153)

k=N

We define the functions φN (λ) =

∞ n=0

and

(2n)

φN λn

(12.154)

Form factor expansions of C(N, N ) and C(0, N ) ∞

FÑ (λ) =

(2n) FÑ λn

¿

(12.155)

n=1

where φN (0) = 1 and FÑ (0) = 0. We will show that ˜

φN (λ) = eFN (λ) ,

(12.156)

from which if we set λ = 1 and use (12.110), we obtain DN /DN +1 = exp

∞

(2k) FÑ .

(12.157)

k=1

Thus it follows from equations (12.106) and (12.153) that DN = (1 − t)

1/4

∞

Dn /Dn+1 = (1 − t)

1/4

exp

= (1 − t)1/4 exp

∞

(2m) F˜k

k=N m=1 ∞

n=N ∞

∞ ∞

(2m) = (1 − t)1/4 exp F˜k

m=1 k=N

(2n)

FN

.

(12.158)

n=1

This proves equations (12.149)–(12.151). It remains to show that equation (12.156) holds. Since φ(0) = 1 and F (0) = 0, equation (12.156) is equivalent to the equation dφ(λ) dF˜ (λ) dF˜ (λ) ˜ = eF (λ) = φ(λ) . dλ dλ dλ

(12.159)

Thus it follows from equations (12.154), (12.155) and (12.159), by equating like powers of λ, that equation (12.156) is equivalent to the following equation: (2n)

nφN

=

n

(2l) (2n−2l)

lFÑ φN

(12.160)

l=1

The left-hand side of (12.160) is (2n) nφN

n+1

= n(−1) n

1 (2π)2n

2n

dzj zjN +1

j=1

−1 −1 P (z2j )P (z2j )Q(z2j−1 )Q(z2j−1 )

j=1

and the right-hand side is

2n−1 j=1

1 1 1 − zj zj+1 z2n z1

(12.161)

¿


n

(2l) (2n−2l)

lFÑ φN

=

l=1

n 2n dzj zjN 1 −1 −1 (−1) P (z2j )P (z2j )Q(z2j−1 )Q(z2j−1 ) (2π)2n 1 − z z j j+1 j=1 j=1 n−1 2l 2n−1 2n 1 (1 − zk )(1 − z2l z2l+1 )(1 − z2n z1 ) zm − (1 − zp ) 1 − z1 z2l p=1 n

l=1

k=1

m=2l+2

(12.162) where the product

2n−1

m=2l+2 zm

2n j=1

is defined as 1 when l = n − 1. The product

n zjN −1 −1 P (z2j )P (z2j )Q(z2j−1 )Q(z2j−1 ) 1 − zj zj+1 j=1

(12.163)

is symmetric both in even and in odd variables separately. Hence 1 − rewritten (under the integration sign ) as 1−

2n

zk ≡ (1 − z1 z2n )(1 +

n−1 2q+1

2n

k=1 zk

zr ).

can be

(12.164)

q=1 r=2

k=1

Next, note that the summand 1 (1 − z2l z2l+1 )(1 − z2n z1 ) 1 − z1 z2l

2n−1

zm

(12.165)

m=2l+2

2l−1 does not involve any of the variables {zi }i=1 . Hence the product 1 −

1−

2l

zk ≡ (1 − z1 z2l )(1 +

l−1 2q+1

2l

k=1 zk

becomes

zr ).

(12.166)

q=1 r=2

k=1

Then the relevant factor of the integrand of the right-hand side of (12.162) becomes n−1 l−1 2q+1 (1 − z2n z1 ) (1 − z2l z2l+1 )(1 + zr ) q=1 r=2

l=1

2n−1 m=2l+2

zm − (1 +

n−1 2q+1

zr )

q=1 r=2

n−1 l−1 2q+1 2n−1 2n−1 n−1 2q+1 (1 + zr )( zm − zm ) − (1 + zr ) . = (1 − z2n z1 ) l=1

q=1 r=2

m=2l+2

m=2l

q=1 r=2

(12.167) After expansion of the first summand this becomes


(1 − z2n z1 )

n−1

2n−1

zm −

n−1 2n−1

zr − (1 +

l=1 r=2

l=1 m=2l+2

n−1 2q+1

¿

zr )

(12.168)

q=1 r=2

under integration which, after summation, reduces to 2n−1

−n(1 − z2n z1 )

zr .

(12.169)

r=2

Thus the equality (12.160) holds, and hence we have proven the desired result (12.149)(12.151). The form factor expansion It remains to show for T < Tc that the exponential form of the correlation (12.149)(12.151) can be rewritten in the form factor representation (10.76): DN = (1 − t)1/4 {1 +

∞

(2n)

fN

}

(12.170)

n=1

with 1 1 = 2 (n!) (2πi)2n

(2n) fN

1≤j≤n 1≤k≤n

2n

dzj zjN

j=1

2

1

1 − z2j−1 z2k

n

−1 −1 Q(z2j−1 )Q(z2j−1 )P (z2j )P (z2j )

j=1

(z2j−1 − z2k−1 )2 (z2j − z2k )2 . (12.171)

1≤j

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